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AMERICAN  HOUSE-CARPENTER : 

A  TREATISE  UPON 

ARCHITECTURE, 

CORNICES  AND  MOULDINGS, 

FRAMING, 

DOORS,  WINDOWS,  AND  STAIRS. 

TOGETHER  WITH 

THE  MOST  IMPORTANT  PRINCIPLES 

OF 

PRACTICAL  GEOMETRY. 


BY  R.  G.  HATFIELD, 

ARCHITECT. 


Jllustratetf  lifl  more  tijan  tfjm  fjunUceO  aEngrabfnQS. 


NEW-YORK  &  LONDON : 

WILEY  AND  PUTNAM. 


1844. 


Entered  aocerding  to  the  Act  of  Congress,  in  the  year  1844,. 

BY  R.  G.  HATFIELD, 

In  the  Clerk’s  office  of  the  District  Court  of  the  Southern  District  of  New-York. 


nkw-tohk: 

WILLIAM  OSBORN,  PRINTER, 
63  WH,LIiM-«TRK*T, 


THE  GETTY  CENTER 

LIBRARY 


PREFACE. 


This  book  is  intended  for  carpenters — for  masters, 
journeymen  and  apprentices.  It  has  long  been  the 
complaint  of  this  class  that  architectural  books,  in¬ 
tended  for  their  instruction,  are  of  a  price  so  high  as 
to  be  placed  beyond  their  reach.  This  is  owing,  in  a 
great  measure,  to  the  costliness  of  the  plates  with 
which  they  are  illustrated :  an  unnecessary  expense,  as 
illustrations  upon  wood,  printed  on  good  paper,  answer 
every  useful  purpose.  Wood  engravings,  too,  can  be 
distributed  among  the  letter-press  ;  an  advantage 
which  plates  but  partially  possess,  and  one  of  great 
importance  to  the  reader. 

Considerations  of  this  kind  induced  the  author  to 
undertake  the  preparation  of  this  volume.  The  sub¬ 
ject  matter  has  been  gleaned  from  works  of  the  first 
authority,  and  subjected  to  the  most  careful  examina¬ 
tion.  The  explanations  have  all  been  written  out 
from  the  figures  themselves,  and  not  taken  from  any 
other  work;  and  the  figures  have  all  been  drawn  ex¬ 
pressly  for  this  book.  In  doing  this,  the  utmost  care 
has  been  taken  to  make  every  thing  as  plain  as  the 
mature  of  the  case  would  admit. 


PREFACE. 


iv 

The  attention  of  the  reader  is  particularly  directed  to 
the  following  new  inventions,  viz  :  an  easy  method  of 
describing  the  curves  of  mouldings  through  three 
given  points  ;  a  rule  to  determine  the  projection  of 
eave  cornices  ;  a  new  method  of  proportioning  a  cor¬ 
nice  to  a  larger  given  one  ;  a  way  to  determine  the 
lengths  and  bevils  of  rafters  for  hip-roofs  ;  a  way  to 
proportion  the  rise  to  the  tread  in  stairs  ;  to  determine 
the  true  position  of  butt-joints  in  hand-rails  ;  to  find 
the  bevils  for  splayed-work  ;  a  general  rule  for  scrolls, 
&c.  Many  problems  in  geometry,  also,  have  been 
simplified,  and  new  ones  introduced.  Much  labour 
has  been  bestowed  upon  the  section  on  stairs,  in  which 
the  subject  of  hand-railing  is  presented,  in  many  re¬ 
spects,  in  a  new,  and,  it  is  hoped,  more  practical  form 
than  in  previous  treatises  on  that  subject. 

The  author  has  endeavoured  to  present  a  fund  of 
useful  information  to  the  American  house-carpenter 
that  would  enable  him  to  excel  in  his  vocation  ;  how 
far  he  has  been  successful  in  that  object,  the  book 
itself  must  determine. 


* 


TABLE  OF  CONTENTS. 


INTRODUCTION. 

Art. 

Articles  necessary  for  drawing,  2  To  use  the  set-square, 
To  prepare  the  paper,  *  5  Directions  for  drawing, 


Art. 

11 
-  13 


SECT.  I.— PRACTICAL  GEOMETRY. 


DEFINITIONS. 


Lines,  -  -  -  17 

Angles,  -  23 

Angular  point,-  27 

Polygons,  -  28 

The  circle,  47 

The  cone,  ...  56 

Conic  sections,  -  -  -  58 

The  ellipsis,  -  -  •  61 

The  cylinder,  68 

PROBLEMS. 

To  bisect  a  line,  -  -  71 


To  erect  a  perpendicular,  -  72 

To  let  fall  a  perpendicular,  73 
To  erect  ditto  on  end  of  line,  74 
Six,  eight  and  ten  rule,  -  74 

To  square  end  of  board,  -  74 

To  square  foundations,  &c.,  74 

To  let  fall  a  perpendicular 
near  the  end  of  a  line,  -  75 

To  make  equal  angles,  -  76 

To  bisect  an  angle,  -  -  77 

To  trisect  a  right  angle,  78 
To  draw  parallel  lines,  -  79 

To  divide  a  line  into  equal 
parts,  -  -  -  80 

To  find  the  centre  of  a  circle,  81 
To  draw  tangent  to  circle,  82 

Do.  without  using  centre,  83 
To  find  the  point  of  contact,  84 
To  draw  a  circle  through  three 
given  points,  -  -  85 


To  find  a  fourth  point  in  circle,  86 
To  describe  a  segment  of  a 
circle  by  a  set-triangle,  -  87 

Do.  by  intersection  of  lines,  88 
To  curve  an  angle,  -  89 

To  inscribe  a  cii’cle  within  a 
given  triangle,  -  90 

To  make  triangle  about  circle,  91 
To  find  the  length  of  a  cir¬ 
cumference,  -  -  92 

To  describe  a  triangle,  hexa¬ 
gon,  &c.,  ...  93 

To  draw  an  octagon,  -  94 

To  eight-square  a  rail,  &c.,  94 

To  describe  any  polygon  in 
a  circle,  ...  95 

To  draw  equilateral  triangle,  96 
To  draw  a  square  by  com¬ 
passes,  -  -  -  97 

To  draw  any  polygon  on  a 
given  line,  98 

To  form  a  triangle  of  required 
size,  -  -  -  .  99 

To  copy  any  right-lined  figure,  100 
To  make  a  parallelogram 

equal  to  a  triangle,  -  101 

To  find  the  area  of  a  triangle,  101 
To  make  one  parallelogram 
equal  another,  -  -  102 

To  make  one  square  equal  to 
two  others,  -  -  -  103 

To  find  the  length  of  a  rafter,  103 


VI 


CONTENTS. 


Art. 

To  find  the  length  of  a  brace,  103 
To  ascertain  the  pitch  of  a 
roof,  -  103 

To  ascertain  the  rake  of  a 
step-ladder,  -  -  -  103 

To  describe  one  circle  equal 
to  two  others,  -  -  104 

To  make  one  polygon  equal 
to  two  or  more,  -  -  104 

To  make  a  square  equal  to 
a  rectangle,  -  -  105 

To  make  a  square  equal  to 
a  triangle,  -  106 

To  find  a  third  proportional,  107 
To  find  a  fourth  proportional,  108 
To  proportion  one  ellipsis  to 
another,  -  108 

To  divide  a  line  as  another,  109 
To  find  a  mean  proportional,  110 
Definitions  of  conic  sections,  111 
To  find  the  axes  of  an  ellipti¬ 


cal  section,  -  -  -  112 

To  find  the  axes  and  base  of 
the  parabola,  -  -  113 

To  find  the  height,  base  and 
axes  of  the  hyperbola,  -  114 

To  find  foci  of  ellipsis,  -  115 

To  describe  an  ellipsis  with 
a  string,  -  -  -  115 

To  describe  an  ellipsis  with 
a  trammel,  -  -  116 

To  construct  a  trammel,  -  116 

To  describe  an  ellipsis  by  or¬ 
dinates,  -  -  -  117 

To  trace  a  curve  through 
given  points,  -  -  117 

To  describe  an  ellipsis  by  in¬ 
tersection  of  lines,  -  118 


Art. 

Do.  from  conjugate  diameters,  118 
Do.  by  intersecting  arcs,  -  119 

To  describe  an  oval  by  com¬ 
passes,  -  -  -  120 

Do.  in  the  proportion,  7  x  9j 
5x7,  &c.,  -  -  -  121 

To  draw  a  tangent  to  an  el¬ 
lipsis,  -  -  -  122 

To  find  the  point  of  contact,  123 
To  find  a  conjugate  to  the 
given  diameter,  -  124 

To  find  the  axes  from  given 
diameters,  -  125 

To  find  axes  proportionate  to 
given  ones,  -  -  126 

To  describe  a  parabola  by  in¬ 
tersection  of  lines,  -  -  127 

To  describe  hyperbola  by  do.,  128 

DEMONSTRATIONS. 

Definitions,  axioms,  &c.,  130.  139 


Addition  of  angles,  -  140 

Equal  triangles,  -  -  141 

Angles  at  base  of  isoceles  tri¬ 
angle  equal,  -  -  142 

Parallelograms  divided  equal¬ 
ly  by  diagonal,  -  -  143 

Equal  parallelograms,  -  144 

Parallelogram  equal  triangles,  146 
To  make  triangle  equal  poly¬ 
gon,  -  147 

Opposite  angles  equal,  -  148 

Angles  of  triangle  equal  two 

right  angles,  -  -  -  149 

Corollaries  from  do.,  150.  155 
Angle  in  semi-circle  a  right 
angle,  -  -  -  156 

Hecatomb  problem,  -  -  157 


SECT.  II.— ARCHITECTURE. 


HISTORY. 

Antiquity  of  its  origin,  -  159 

Its  cultivation  among  the  an¬ 
cients,  -  -  -  160 

Among  the  Greeks,  *  -  161 


Among  the  Romans,  -  162 

Ruin  caused  by  Goths  and 
Vandals,  -  -  -  163 

Of  the  Gothic,  -  -  164 

Of  the  Lombard,  -  -  165 


CONTENTS. 


VII 


Art. 

Ofthe  Byzantine  and  Oriental,  166 
Moorish,  Arabian  and  Modern 
Gothic,  167 

Of  the  English,  -  -  168 

Revival  of  the  art  in  the  sixth 
century,  -  -  -  169 

The  art  improved  in  the  14th 
and  15th  centuries,  -  170 

Roman  styles  cultivated,  171 

STYLES. 

Origin  of  different  styles,  172 
Stylobate  and  pedestal,  -  173 

Definitions  of  an  order,  -  174 

Of  the  several  parts  of  an 
order,  -  -  175.  185 

To  proportion  an  order,  -  186 

The  Grecian  orders,  -  187 

Origin  of  the  Doric,  -  -  188 

Intercolumniation,  -  -  189 

Adaptation,  -  -  -  190 

Origin  of  the  Ionic,  -  191 

Characteristics,  -  -  192 

Intercolumniation,  -  -  193 

Adaptation,  -  -  -  194 

To  describe  the  volute,  -  195 

Origin  of  the  Corinthian,  -  196 

Adaptation,  -  -  -  197 

Persians,  -  -  -  -  199 

Caryatides,  -  -  -  200 

The  Roman  orders,  -  -  202 


Art. 

Extent  of  Roman  structures,  202 
Roman  styles  copied  from 

Grecian,  -  -  -  203 

Origin  of  the  Tuscan,  -  204 

Adaptation,  -  -  -  205 

Characteristics  of  the  Egypt¬ 
ian,  ....  206 

Extent  of  Egyptian  structures,  206 
Adaptation,  -  207 

Appropriateness ofdesign,  208.  211 
Durable  structures,  -  -  212 

Plans  of  dwellings,  &c.,  213 

Directions  for  designing,  213,  214 

PRINCIPLES. 

Origin  of  the  art,  -  -  215 

Arrangement  and  design,  -  216 

Ventilation  and  cleanliness,  217 
Stability,  -  -  -  218 

Ornaments,  -  219 

Scientific  knowledge  neces¬ 
sary,  -  -  -  220 

The  foundation,  -  -  221 

The  column,  -  -  -  222 

The  wall,  -  -  -  223 

The  lintel,  -  -  -  224 

The  arch,  -  -  -  225 

The  vault,  ...  226 

The  dome,  ...  227 

The  roof,  -  -  -  228 


SECT.  III.— MOULDINGS,  CORNICES,  &c. 


MOULDINGS,  &C. 

Elementary  forms,  -  -  229 

Characteristics,  -  -  230 

Grecian  and  Roman,  -  -  231 

Profile,  -  -  -  232 

To  describe  the  torus  and 
scotia,  -  233 

To  describe  the  echinus,  234 
To  describe  the  cavetto,  235 
To  describe  the  cyma-recta,  236 
To  describe  the  cyma-reversa,  237 


Roman  mouldings, 

- 

238' 

Modern  mouldings, 

- 

239 

Antes  caps, 

- 

240 

CORNICES. 

Designs,  - 

- 

241 

To  proportion  an  eave  cornice, 

242 

Do.  from  a  smaller 

given 

one,  - 

Do.  from  a  larger 

- 

243 

given 

one, 

- 

244 

To  find  shape  of  angle-bracket,  245 
To  find  form  of  raking  cornice,  246 


Vlll 


CONTENTS. 


SECT. 

Laws  of  pressure, 
Parallelogram  of  forces,  - 
To  measure  the  pressure  on 
rafters,  - 
Do.  on  tie-beams, 

The  effect  of  position, 

The  composition  of  forces, 
Best  position  for  a  strut,  - 
Nature  of  ties  and  struts, 

To  distinguish  ties  from  struts, 
Lattice-work  framing, 
Direction  of  pressure  in  raft¬ 
ers,  - 

Oblique  thrust  of  lean-to  roofs, 
Pressure  on  floor-beams,  - 
Kinds  of  pressure, 

Resistance  to  compression, 
Area  of  post, 

Resistance  to  tension, 

Area  of  suspending  piece, 
Resistance  to  cross-strains, 
Area  of  bearing  timbers, 
Area  of  stiffest  beam, 

Bearers  narrow  and  deep, 
Principles  of  framing, 

FLOORS. 

Single-joisted, 

To  find  area  of  floor-timbers, 
Dimensions  of  trimmers,  &c., 
Strutting  between  beams, 
Cross-furring  and  deafening, 
Double  floors,  - 
Dimensions  of  binding-joists, 
Do.  of  bridging-joists, 

Do.  of  ceiling-joists,  - 
Framed  floors,  - 
Dimensions  of  girders,  - 
Girders  sawn  and  bolted,  - 
Trussed  girders, 

Floors  in  general, 

PARTITIONS. 

Nature  of  their  construction, 
Designs  for  partitions, 
Superfluous  timber,  - 
Improved  method,  - 
Weight  of  partitioning, 


-FRAMING. 

ROOFS. 

Art. 

Lateral  strains,  -  .  285 

Pressure  on  roofs,  -  -  286 

Weight  of  covering,  -  286 

Definitions,  ...  287 

Relative  size  of  timbers,  288 
To  find  the  area  of  a  king-post,  289 
Of  a  queen-post,  -  -  290 

Of  a  tie-beam,  -  -  -  291 

Of  a  rafter,  -  -  -  292 

Of  a  straining-beam,  -  294 

Of  braces,  -  295 

Of  purlins,  ...  296 

Of  common  rafters,  -  297 

To  avoid  shrinkage,  -  -  298 

Roof  with  a  built-rib,  -  299 

Badlv-constructed  roofs,  -  300 

To  find  the  length  and  bevils 
in  hip-roofs,  -  -  301 

To  find  the  backing  of  a  hip- 
rafter,  ...  -  302 

DOMES. 

With  horizontal  ties,  -  303 

Ribbed  dome,  ...  304 
Area  of  the  ribs,  -  *  305 

Curve  of  equilibrium,  -  306 

To  describe  a  cubic  parabola,  307 
Small  domes  for  stairways,  308 
To  find  the  curves  of  the  ribs,  309 
To  find  the  shape  of  the  cover¬ 
ing  for  spherical  domes,  310 
Do.  when  laid  horizontally,  311 
To  find  an  angle-rib,  -  -  312 

BRIDGES. 

Wooden  bridge  with  tie-beam,  313 
Do.  without  a  tie-beam,  314 
Do.  with  a  built-rib,  315 
Table  of  least  rise  in  bridges,  315 
Rule  for  built-ribs,  -  -  315 

Pressure  on  arches,  -  316 

To  form  bent- ribs,  -  -  317 

Elasticity  of  timber,  -  317 

To  construct  a  framed  rib,  318 
Width  of  roadway,  &c.,  -  319 

Stone  abutments  and  piers,  320 
Piers  constructed  of  piles,  321 


IV.- 

Art. 

248 

248 

249 

250 

251 

252 

253 

254 

255 

256 

257 

258 

259 

260 

261 

261 

262 

262 

263 

263 

264 

265 

266 

267 

268 

269 

270 

271 

272 

273 

274 

275 

276 

277 

278 

279 

280 

281 

282 

282 

283 

284 


CONTENTS'. 


IX 


Piles  in  ancient  bridges,  321 

Centring  for  stone  bridges,  322 

Pressure  of  arch-stones,  -  322 

Centre  without  a  tie  at  the 
base,  -  -  -  323 

Construction  of  centres,  -  324 

General  directions,  -  325 

Lowering  centres,  -  -  326 

Relative  size  of  timbers,  -  327 

Short  rule  for  do.  -  -  328 

Joints  between  arch-stones,  329 

Do.  in  elliptical  arch,  -  330 

Do.  in  parabolic  arch,  -  331 


JOINTS. 

Art. 

Scarfing,  or  splicing,  332.  334 
To  proportion  the  parts,  -  335 

Joints  in  beams  and  posts,  -  336 

Joints  in  floor-timbers,  -  337 

Timber  weakened  by  framing,  338 
Joints  for  rafters  and  braces,  339 
Evil  of  shrinking  avoided,  -  340 

Proper  joint  for  collar-beam,  341 
Pins,  nails,  bolts  and  straps,  342 
Dimensions  of  straps,  -  342 

To  prevent  the  rusting  of 
straps,  -  -  -  -  342 


SECT.  V.— DOORS,  WINDOWS,  &c. 


DOORS. 

Dimensions  of  doors,  -  -  343 

To  proportion  height  to  width,  344 
Width  of  stiles,  rails  and 
panels,  ...  345 

Example  of  trimming,  -  346 

Elevation  of  a  door  and  trim¬ 
mings,  -  347 

General  directions  for  hang¬ 
ing  doors,  -  -  -  348 


WINDOWS. 

To  determine  the  size,  -  34£ 

To  find  dimensions  of  frame,  350 
To  proportion  box  to  flap 
shutter,  -  351 

To  proportion  and  arrange 
windows,  -  -  -  352 

Circular-headed  windows,  353 
To  find  the  fofrn  of  the  soffit,  354 
Do.  in  a  circular  wall,  -  355 


SECT.  VI. 

Their  position,  -  -  -  356 

Principles  of  the  pitch-board,  357 
To  proportion  the  rise  to  the 
tread,  -  -  -  358 

The  angle  of  ascent,  -  -  359 

Length  of  steps,  -  -  360 

To  construct  a  pitch-board,  361 
To  lay-out  the  string,  -  362 

Section  of  step,  -  -  363 

PLATFORM  STAIRS. 

To  construct  the  cylinder,  -  364 

To  cut  the  lower  edge  of  do.,  365 
To  place  the  balusters,  -  366 


■STAIRS. 


To  find  the  moulds  for  the 
rail,  -  -  -  -  367 

Elucidation  of  this  method,  368 
Two  other  examples,  369,  370 
To  apply  the  mould  to  the 
plank,  -  -  -  371 

To  bore  for  the  balusters,  -  372 


Face-mould  for  moulded  rail,  373 
To  apply  this  mould  to  plank,  374 
To  ascertain  thickness  of  stuff,  375 

WINDING  STAIRS. 

Flyers  and  winders,  -  376 

To  construct  winding  stairs,  377 


X 


CONTENTS. 


Art. 

Timbers  to  support  winding 
stairs,  -  -  -  -  378 

To  find  falling-mould  of  rail,  379 
To  find  face-mould  of  do.,  380 

Position  of  butt-joint,  -  380 

To  ascertain  thickness  of 
stuff,  -  381 

To  apply  the  mould  to  plank,  383 
Elucidation  of  the  butt-joint,  384 
Quarter-circle  stairs,  -  385 

Falling-mould  for  do.,  -  386 

Face-mould  for  do.,  -  387 

Elucidation  of  this  method,  388 
To  bevil  edge  of  plank,  -  389 

To  apply  moulds  without  be- 
villing  plank,  -  -  390 


Art 

To  find  bevils  for  splayed- 
work,  -  -  -  391 

Another  method  for  face- 

moulds,  -  392 

To  apply  face-mould  to  plank,  394 
To  apply  falling-mould,  -  395 

SCROLLS. 

General  rule,  -  -  396 

To  describe  scroll  for  rail,  398 
For  curtail-step,  -  -  399 

Balusters  under  scroll,  -  400 

Falling-mould  for  scroll,  -  401 

Face-mould  for  do.,  -  402 

Round  rails  over  winders,  -  403 

To  find  form  of  newel-cap,  404 


APPENDIX. 


Glossary  of  Architectural  Terms, 

Table  of  Squares,  Cubes  and  Roots, 

Rules  for  extending  the  use  of  the  foregoing  table, 

Rule  for  finding  the  roots  of  whole  numbers  with  decimals, 
Rules  for  the  reduction  of  Decimals, 

Table  of  Areas  and  Circumferences  of  Circles,  - 
Rules  for  extending  the  use  of  the  foregoing  table, 

Table  showing  the  Capacity  of  Wells,  Cisterns,  &c., 

Rules  for  finding  the  Areas,  &c.,  of  Polygons, 

Table  of  Weights  of  Materials,  - 


Page. 

3 

14 

21 

23 

23 

25 

28 

29 

39 

31 


INTRODUCTION. 


Art.  1. — A  knowledge  of  the  properties  and  principles  of  lines 
can  best  be  acquired  by  practice.  Although  the  various  problems 
throughout  this  work  may  be  understood  by  inspection,  yet  they 
will  be  impressed  upon  the  mind  with  much  greater  force,  if  they 
are  actually  performed  with  pencil  and  paper  by  the  student. 
Science  is  acquired  by  study — art  by  practice  :  he,  therefore,  who 
would  have  any  thing  more  than  a  theoretical,  (which  must  of 
necessity  be  a  superficial,)  knowledge  of  Carpentry,  will  attend 
to  the  following  directions,  provide  himself  with  the  articles  here 
specified,  and  perform  all  the  operations  described  in  the  follow¬ 
ing  pages.  Many  of  the  problems  may  appear,  at  the  first  read¬ 
ing,  somewhat  confused  and  intricate  ;  but  by  making  one  line 
at  a  time,  according  to  the  explanations,  the  student  will  not 
only  succeed  in  copying  the  figures  correctly,  but  by  ordinary 
attention  will  learn  the  principles  upon  which  they  are  based, 
and  thus  be  able  to  make  them  available  in  any  unexpected  case 
to  which  they  may  apply. 

2.  — The  following  articles  are  necessary  for  drawing,  viz  :  a 
drawing-board,  paper,  drawing-pins  or  mouth-glue,  a  sponge,  a 
T-square,  a  set-square,  two  straight-edges,  or  flat  rulers,  a  lead 
pencil,  a  piece  of  india-rubber,  a  cake  of  india-ink,  a  set  of  draw¬ 
ing-instruments,  and  a  scale  of  equal  parts. 

3.  — The  size  of  the  drawing-board  must  be  regulated  accord¬ 
ing  to  the  size  of  the  drawings  which  are  to  be  made  upon  it. 
Yet  for  ordinary  practice,  in  learning  to  draw,  a  board  about  15 

1 


. 

2  AMERICAN  HOUSE  CARPENTER. 

by  20  inches,  and  one  inch  thick,  will  be  found  large  enough, 
and  more  convenient  than  a  larger  one.  This  board  should  be 
well-seasoned,  perfectly  square  at  the  corners,  and  without 
clamps  on  the  ends.  A  board  is  better  without  clamps,  because 
the  little  service  they  are  supposed  to  render  by  preventing  the 
board  from  warping,  is  overbalanced  by  the  consideration  that 
the  shrinking  of  the  panel  leaves  the  ends  of  the  clamps  project¬ 
ing  beyond  the  edge  of  the  board,  and  thus  interfering  with  the 
proper  working  of  the  stock  of  the  T-square.  When  the  stuff 
is  well-seasoned,  the  warping  of  the  board  will  be  but  trifling  ; 
and  by  exposing  the  rounding  side  to  the  fire,  or  to  the  sun,  it 
may  be  brought  back  to  its  proper  shape. 

4.  — For  mere  line  drawings,  the  paper  need  not  commonly 
be  what  is  called  drawing-paper ;  as  this  is  rather  costly,  and 
will,  where  much  is  used,  make  quite  an  item  of  expense. 
Cartridge-paper,  as  it  is  called,  of  about  20  by  26  inches,  and  of 
as  good  a  quality  nearly  as  drawing-paper,  can  be  bought  for 
about  50  cts.  a  quire,  or  2  pence  a  sheet ;  and  each  sheet  may  be 
cut  in  halves,  or  even  quarters,  for  practising.  If  the  drawing 
is  to  be  much  used,  as  working  drawings  generally  are,  cartridge- 
paper  is  much  better  than  the  other  kind. 

5.  — A  drawing-pin  is  a  small  brass  button,  having  a  steel  pin 
projecting  from  the  under  side.  By  having  one  of  these  at  each 
corner,  the  paper  can  be  fixed  to  the  board  ;  but  this  can  be  done 
in  a  much  better  manner  with  moutli-glne.  The  pins  will  pre¬ 
vent  the  paper  from  changing  its  position  on  the  board  ;  but, 
more  than  this,  the  glue  keeps  the  paper  perfectly  tight  and 
smooth,  thus  making  it  so  much  the  more  pleasant  to  work  on. 

To  attach  the  paper  with  mouth-glue,  lay  it  with  the  bottom 
side  up,  on  the  board ;  and  with  a  straight-edge  and  penknife, 
cut  off  the  rough  and  uneven  edge.  With  a  sponge  moderately 
wet,  rub  all  the  surface  of  the  paper,  except  a  strip  around  the 
edge  about  half  an  inch  wide.  As  soon  as  the  glistening  of  the 
water  disappears,  turn  the  sheet  over,  and  place  it  upon  the 


INTRODUCTION. 


3 


board  just  where  you  wish  it  glued.  Commence  upon  one  of 
the  longest  sides,  and  proceed  thus :  lay  a  flat  ruler  upon  the 
paper,  parallel  to  the  edge,  and  within  a  quarter  of  an  inch  of  it. 
With  a  knife,  or  any  thing  similar,  turn  up  the  edge  of  the  paper 
against  the  edge  of  the  ruler,  and  put  one  end  of  the  cake  of 
mouth-glue  between  your  lips  to  dampen  it.  Then  holding  it 
upright,  rub  it  against  and  along  the  entire  edge  of  the  paper 
that  is  turned  up  against  the  ruler,  bearing  moderately  against 
the  edge  of  the  ruler,  which  must  be  held  firmly  with  the  left 
hand.  Moisten  the  glue  as  often  as  it  becomes  dry,  until  a 
sufficiency  of  it  is  rubbed  on  the  edge  of  the  paper.  Take 
away  the  ruler,  restore  the  turned-up  edge  to  the  level  of  the 
board,  and  lay  upon  it  a  strip  of  pretty  stiff  paper.  By  rubbing 
upon  this,  not  very  hard  but  pretty  rapidly,  with  the  thumb  nail 
of  the  right  hand,  so  as  to  cause  a  gentle  friction,  and  heat  to  be 
imparted  to  the  glue  that  is  on  the  edge  of  the  paper,  you  will 
make  it  adhere  to  the  board.  The  other  edges  in  succession 
must  be  treated  in  the  same  manner. 

Some  short  distances  along  one  or  more  of  the  edges,  may 
afterwards  be  found  loose :  if  so,  the  glue  must  again  be  applied, 
and  the  paper  rubbed  until  it  adheres.  The  board  must  then  be 
laid  away  in  a  warm  or  dry  place ;  and  in  a  short  time,  the  sur¬ 
face  of  the  paper  will  be  drawn  out,  perfectly  tight  and  smooth, 
and  ready  for  use.  The  paper  dries  best  when  the  board  is  laid 
level.  When  the  drawing  is  finished,  lay  a  straight-edge  upon 
the  paper,  and  cut  it  from  the  board,  leaving  the  glued  strip  still 
attached.  This  may  afterwards  be  taken  off  by  wetting  it  freely 
with  the  sponge ;  which  will  soak  the  glue,  and  loosen  the 
paper.  Do  this  as  soon  as  the  drawing  is  taken  off,  in  order  that 
the  board  may  be  dry  when  it  is  wanted  for  use  again.  Care 
must  be  taken  that,  in  applying  the  glue,  the  edge  of  the  paper 
does  not  become  damper  than  the  rest :  if  it  should,  the  paper 
must  be  laid  aside  to  dry,  (to  use  at  another  time,)  and  another 
sheet  be  used  in  its  place. 


4 


AMERICAN  HOUSE  CARPENTER 


Sometimes,  especially  when  the  drawing  hoard  is  new,  the 
paper  will  not  stick  very  readily  ;  but  by  persevering,  this  diffi¬ 
culty  may  be  overcome.  In  the  place  of  the  mouth-glue,  a 
strong  solution  of  gum-arabic  may  be  used,  and  on  some 
accounts  is  to  be  preferred ;  for  the  edges  of  the  paper  need  not 
be  kept  dry,  and  it  adheres  more  readily.  Dissolve  the  gum  in 
a  sufficiency  of  warm  water  to  make  it  of  the  consistency  of 
linseed  oil.  It  must  be  applied  to  the  paper  with  a  brush,  when 
the  edge  is  turned  up  against  the  ruler,  as  was  described  for  the 
mouth-glue.  If  two  drawing-boards  are  used,  one  may  be  in  use 
while  the  other  is  laid  away  to  dry  ;  and  as  they  may  be  cheaply 
made,  it  is  advisable  to  have  two.  The  drawing-board  having 
a  frame  around  it,  commonly  called  a  panel-board,  may  afford 
rather  more  facility  in  attaching  the  paper  when  this  is  of  the 
size  to  suit ;  yet  it  has  objections  which  overbalance  that  con¬ 
sideration. 

6. — A  T-square  of  mahogany,  at  once  simple  in  its  construc¬ 
tion,  and  affording  all  necessary  service,  may  be  thus  made. 
Let  the  stock  or  handle  be  seven  inches  long,  two  and  a  quarter 
inches  wide,  and  three-eighths  of  an  inch  thick:  the  blade, 
twenty  inches  long,  (exclusive  of  the  stock,)  two  inches  wide, 
and  one-eighth  of  an  inch  thick.  In  joining  the  blade  to  the 
stock,  a  very  firm  and  simple  joint  may  be  made  by  dovetailing 
it — as  shown  at  Fig.  1. 


Fig.  I. 


INTRODUCTION. 


5 


7.  — The  set-square  is  in  the  form  of  a  right-angled  triangle  ; 

and  is  commonly  made  of  mahogany,  one-eighth  of  an  inch  in 
thickness.  The  size  that  is  most  convenient  for  general  use,  is 
six  inches  and  three  inches  respectively  for  the  sides  which  con¬ 
tain  the  right  angle ;  although  a  particular  length  for  the  sides  is 
by  no  means  necessary.  Care  should  be  taken  to  have  the  square 
corner  exactly  true.  This,  as  also  the  T-square  and  rulers, 
should  have  a  hole  bored  through  them,  by  which  to  hang  them 
upon  a  nail  when  not  in  use.  ^ 

8.  — One  of  the  rulers  may  be  about  twenty  inches  long,  and 
the  other  six  inches.  The  pencil  ought  to  be  hard  enough  to 
retain  a  fine  point,  and  yet  not  so  hard  as  to  leave  ineffaceable 
marks.  It  should  be  used  lightly,  so  that  the  extra  marks  that 
are  not  needed  when  the  drawing  is  inked,  may  be  easily  rubbed 
off  with  the  rubber.  The  best  kind  of  india-infc  is  that  which 
will  easily  rub  off  upon  the  plate  ;  and,  when  the  cake  is  rub¬ 
bed  against  the  teeth,  will  be  free  from  grit. 

9.  — The  drawing-instrumeyits  may  be  purchased  of  mathe¬ 
matical  instrument  makers  at  various  prices  :  from  one  to  one 
hundred  dollars  a  set.  In  choosing  a  set,  remember  that  the 
lowest  price  articles  are  not  always  the  cheapest.  A  set,  com¬ 
prising  a  sufficient  number  of  instruments  for  ordinary  use,  well 
made  and  fitted  in  a  mahogany  box,  may  be  purchased  at  Pike 
and  Son’s,  (Broadway,  near  Maiden-lane,  N.  Y.,)  for  three  or  four 
dollars.  The  compasses  in  this  set  have  a  needle  point,  which 
is  much  preferable  to  a  common  point. 

10. — The  best  scale  of  equal  parts  for  carpenters’  use,  is  one 
that  has  one-eighth,  three-sixteenths,  one-fourth,  three-eighths, 
one-half,  five-eighths,  three-fourths,  and  seven-eighths  of  an 
inch,  and  one  inch,  severally  divided  into  twelfths ,  instead  of 
being  divided,  as  they  usually  are,  into  tenths.  By  this,  if  it  be 
required  to  proportion  a  drawing  so  that  every  foot  of  the  object 
represented  will  upon  the  paper  measure  one-fourth  of  an  inch, 
use  that  part  of  the  scale  which  is  divided  into  one-fourths  of  an 


6 


AMERICAN  HOUSE-CARPENTER. 


inch,  taking  for  every  foot  one  of  those  divisions,  and  for  every 
inch  one  of  the  subdivisions  into  twelfths  ;  and  proceed  in  like 
manner  in  proportioning  a  drawing  to  any  of  the  other  divisions 
of  the  scale.  An  instrument  in  the  form  of  a  semi-circle,  called  a 
' 'protractor ,  and  used  for  laying  down  and  measuring  angles,  is 
of  much  service  to  surveyors,  but  not  much  to  carpenters. 

11. — In  drawing  parallel  lines,  when  they  are  to  be  parallel 
to  either  side  of  the  board,  use  the  T-square ;  but  when  it  is 
required  to  draw  lines  parallel  to  a  line  which  is  drawn  in  a 
direction  oblique  to  either  side  of  the  board,  the  set-square  must 
be  used.  Let  a  b,  {Fig.  2,)  be  a  line,  parallel  to  which  it  is 

b 


desired  to  draw  one  or  more  lines.  Place  any  edge,  as  c  d ,  of 
the  set-square  even  with  said  line ;  then  place  the  ruler,  g  h, 
against  one  of  the  other  sides,  as  c  e,  and  hold  it  firmly ;  slide 
the  set-square  along  the  edge  of  the  ruler  as  far  as  it  is  desired, 
as  at  / ;  and  a  line  drawn  by  the  edge,  i  /,  will  be  parallel  to  a  b. 

12. — To  draw  a  line,  as  k  l ,  {Fig.  3,)  perpendicular  to  another, 
as  a  6,  set  the  shortest  edge  of  the  set-square  at  the  line,  a  b  ; 
place  the  ruler  against  the  longest  side,  (the  hypothenuse  of  the 
right-angled  triangle  ;)  hold  the  ruler  firmly,  and  slide  the  set- 
square  along  until  the  side,  e  d,  touches  the  point,  k  ;  then  the 
line,  Ik,  drawn  by  it,  will  be  perpendicular  to  a  b.  In  like 


INTRODUCTION. 


7 


manner,  the  drawing  of  other  problems  may  be  facilitated,  as  will 
be  discovered  in  using  the  instruments. 


i 


13. — In  drawing  a  problem,  proceed,  with  the  pencil  sharpened 
to  a  point,  to  lay  down  the  several  lines  until  the  whole  figure  is 
completed ;  observing  to  let  the  lines  cross  each  other  at  the 
several  angles,  instead  of  merely  meeting.  By  this,  the  length 
of  every  line  will  be  clearly  defined.  With  a  drop  or  two  of 
water,  rub  one  end  of  the  cake  of  ink  upon  a  plate  or  saucer, 
until  a  sufficiency  adheres  to  it.  Be  careful  to  dry  the  cake  of 
ink ;  because  if  it  is  left  wet,  it  will  crack  and  crumble  in  pieces. 
With  an  inferior  camel’s-hair  pencil,  add  a  little  water  to  the 
ink  that  was  rubbed  on  the  plate,  and  mix  it  well.  It  should  be 
diluted  sufficiently  to  flow  freely  from  the  pen,  and  yet  be  thick 
enough  to  make  a  black  line.  With  the  hair  pencil,  place  a 
little  of  the  ink  between  the  nibs  of  the  drawing-pen,  and  screw 
the  nibs  together  until  the  pen  makes  a  fine  line.  Beginning 
with  the  curved  lines,  proceed  to  ink  all  the  lines  of  the  figure ; 
being  careful  now  to  make  every  line  of  its  requisite  length.  If 
they  are  a  trifle  too  short  or  too  long,  the  drawing  will  have  a 
ragged  appearance ;  and  this  is  opposed  to  that  neatness  and 
accuracy  which  is  indispensable  to  a  good  drawing.  When  the 
ink  is  dry,  efface  the  pencil-marks  with  the  india-rubber.  If 


8 


AMERICAN  HOUSE-CARPENTER. 


the  pencil  is  used  lightly,  they  will  all  mb  off,  leaving  those 
lines  only  that  were  inked. 

14. — In  problems,  all  auxiliary  lines  are  drawn  light ;  while 
the  lines  given  and  those  sought,  in  order  to  be  distinguished  at 
a  glance,  are  made  much  heavier.  The  heavy  lines  are  made 
so,  by  passing  over  them  a  second  time,  having  the  nibs  of  the 
pen  separated  far  enough  to  make  the  lines  as  heavy  as  desired. 
If  the  heavy  lines  are  made  before  the  drawing  is  cleaned  with 
the  rubber,  they  will  not  appear  so  black  and  neat ;  because  the 
india-rubber  takes  away  part  of  the  ink.  If  the  drawing  is  a 
ground-plan  or  elevation  of  a  house,  the  shade-lines,  as  they  are 
termed,  should  not  be  put  in  until  the  drawing  is  shaded ;  as 
there  is  danger  of  the  heavy  lines  spreading,  when  the  brush,  in 
shading  or  coloring,  passes  over  them.  If  the  lines  are  inked 
with  common  writing-ink,  they  will,  however  fine  they  may  be 
made,  be  subject  to  the  same  evil ;  for  which  reason,  india-ink 
is  the  only  kind  to  be  used. 


THE 


AMERICAN  HOUSE-CARPENTER. 


SECTION  I.— PRACTICAL  GEOMETRY. 


DEFINITIONS. 


15.  —  Geometry  treats  of  the  properties  of  magnitudes. 

16.  — A  'point  has  neither  length,  breadth,  nor  thickness. 

17.  — A  line  has  length  only. 

18.  — Superficies  has  length  and  breadth  only. 

19.  — A  plane  is  a  surface,  perfectly  straight  and  even  in  every 
direction  ;  as  the  face  of  a  panel  when  not  warped  nor  winding. 

20.  — A  solid  has  length,  breadth  and  thickness. 

21.  — A  right ,  or  straight ,  line  is  the  shortest  that  can  be 
drawn  between  two  points. 

22.  — Parallel  lines  are  equi-distant  throughout  their  length. 

23.  — An  angle  is  the  inclination  of  two  lines  towards  one 
another.  {Fig.  4.) 


a 


Fig.  4. 


Fig.  5. 

2 


Fig.  & 


10 


AMERICAN  HOUSE-CARPENTER. 


24.  — A  right  angle  has  one  line  perpendicular  to  the  other. 
(Fig.  5.) 

25.  — An  oblique  angle  is  either  greater  or  less  than  a  right 
angle.  (Fig.  4  and  6.) 

26.  — An  acute  angle  is  less  than  a  right  angle.  (Fig-  4.) 

27.  — An  obtuse  angle  is  greater  than  a  right  angle.  (Fig.  6.) 

When  an  angle  is  denoted  by  three  letters,  the  middle  one,  in 
the  order  they  stand,  denotes  the  angular  point,  and  the  other 
two  the  sides  containing  the  angle  ;  thus,  let  a  b  c,  (Fig.  4,)  be 
the  angle,  then  b  will  be  the  angular  point,  and  a  b  and  b  c  will 
be  the  two  sides  containing  that  angle. 


28. — A  triangle  is  a  superficies  having  three  sides  and  angles. 
(Fig.  7,  8,  9  and  10.) 


Fig.  7. 


Fig.  8. 


29. — An  equi-lateral  triangle  has  its  three  sides  equal. 


30.  — An  isoceles  triangle  has  only  two  sides  equal.  (Fig.  8.) 

31.  — A  scalene  triangle  has  all  its  sides  unequal.  (Fig.  9) 


32.  — A  right-angled  triangle  has  one  right  angle.  (Fig.  10.) 

33.  — An  acute-angled  triangle  has  all  its  angles  acute. 
(Fig.  7  and  8.) 

34.  — An  obtuse-angled  triangle  has  one  obtuse  angle. 


(Fig.  9.) 

35. — A  quadrangle  has  four  sides  and  four  angles.  (Fig.  11 
to  16.) 


PRACTICAL  GEOMETRY. 


11 


36.  — A  parallelogram  is  a  quadrangle  having  its  opposite 
sides  parallel.  {Fig.  11  to  14.) 

37.  — A  rectangle  is  a  parallelogram,  its  angles  being  right 
angles.  {Fig.  11  and  12.) 

38.  — A.  square  is  a  rectangle  having  equal  sides.  {Fig.  11.) 

39.  — A  rhombus  is  an  equi-lateral  parallelogram  having  ob¬ 
lique  angles.  {Fig.  13.) 


ZZ7 

Fig.  ,13.  Fig.  14. 

40.  — A  rhomboid  is  a  parallelogram  having  oblique  angles. 
{Fig.  14.) 

41.  — A  trapezoid  is  a  quadrangle  having  only  two  of  its  sides 
parallel.  {Fig.  15.) 


Fig.  15.  Fig.  16. 


42.  — A  trapezium  is  a  quadrangle  which  has  no  two  of  its 
sides  parallel.  {Fig.  16.) 

43.  — A  polygon  is  a  figure  bounded  by  right  lines. 

44.  — A  regular  polygon  has  its  sides  and  angles  equal. 

45.  — An  irregular  polygon  has  its  sides  and  angles  unequal. 

46.  — A  trigon  is  a  polygon  of  three  sides,  {Fig.  7  to  10 ;) 
a  tetragon  has  four  sides,  {Fig.  11  to  16 ;)  a  pentagon  has 


12 


AMERICAN  HOUSE-CARPENTER. 


five,  {Fig.  17  ;)  a  hexagon  six,  {Fig.  18 ;)  a  heptagon  seven, 
{Fig.  19 ;)  an  octagon  eight,  {Fig.  20 ;)  a  nonagon  nine ;  a 
decagon  ten  ;  an  undecagon  eleven  ;  and  a  dodecagon  twelve 
sides. 


47 .  — A  circle  is  a  figure  bounded  by  a  curved  line,  called  the 
circumference  ;  which  is  every  where  equi-distant  from  a  cer¬ 
tain  point  within,  called  its  centre. 

The  circumference  is  also  called  the  periphery,  and  sometimes 
the  circle. 

48.  — The  radius  of  a  circle  is  a  right  line  drawn  from  the 
centre  to  any  point  in  the  circumference,  {a  b,  Fig.  21.) 

All  the  radii  of  a  circle  are  equal. 


e 


49.  — The  diameter  is  a  right  line  passing  through  the  centre, 
and  terminating  at  two  opposite  points  in  the  circumference. 
Hence  it  is  twice  the  length  of  the  radius,  (c  d,  Fig.  21.) 

50.  — An  arc  of  a  circle  is  a  part  of  the  circumference,  (c  b,  or 
bed,  Fig.  21.) 

51.  — A  chord  is  a  right  line  joining  the  extremities  of  an  arc. 
{b  d,  Fig.  21.) 


PRACTICAL  GEOMETRY. 


13 


52.  — A  segment  is  any  part  of  a  circle  bounded  by  an  arc  and 
its  chord.  (A,  Fig.  21.) 

53.  — A  sector  is  any  part  of  a  circle  bounded  by  an  arc  and 
two  radii,  drawn  to  its  extremities.  ( B ,  Fig.  21.) 

54.  — A  quadrant ,  or  quarter  of  a  circle,  is  a  sector  having  a 
quarter  of  the  circumference  for  its  arc.  ((7,  Fig.  21.) 

55.  — A  tangent  is  a  right  line,  which  in  passing  a  curve, 
touches,  without  cutting  it.  (/  g ,  Fig.  21.) 

56.  — A  cone  is  a  solid  figure  standing  upon  a  circular  base 
diminishing  in  straight  lines  to  a  point  at  the  top,  called  its 
vertex.  (Fig.  22.) 


57.  — The  axis  of  a  cone  is  a  right  line  passing  through  it,  from 
the  vertex  to  the  centre  of  the  circle  at  the  base. 

58.  — An  ellipsis  is  described  if  a  cone  be  cut  by  a  plane,  not 
parallel  to  its  base,  passing  quite  through  the  curved  surface. 
(a  b,  Fig.  23.) 

59.  — A  parabola  is  described  if  a  cone  be  cut  by  a  plane, 
parallel  to  a  plane  touching  the  curved  surface,  (c  d ,  Fig.  23 — 
c  d  being  parallel  to  f  g.) 

60.  — An  hyperbola  is  described  if  a  cone  be  cut  by  a  plane, 
parallel  to  any  plane  within  the  cone  that  passes  through  its 
vertex,  (e  h,  Fig.  23.) 

61.  — Foci  are  the  points  at  which  the  pins  are  placed  in  de¬ 
scribing  an  ellipse.  (See  Art.  115,  and /,  /,  Fig.  24.) 


14 


AMERICAN  HOUSE-CARPENTER. 


62.  — The  transverse  axis  is  the  longest  diameter  of  the 
ellipsis.  ( a  b ,  Fig.  24.) 

63.  — The  conjugate  axis  is  the  shortest  diameter  of  the 
ellipsis  ;  and  is,  therefore,  at  right  angles  to  the  transverse  axis, 
(c  d,  Fig.  24.) 

64.  — The  parameter  is  a  right  line  passing  through  the  focus 
of  an  ellipsis,  at  right  angles  to  the  transverse  axis,  and  termina¬ 
ted  by  the  curve,  (g  h  and  g  t ,  Fig.  24.) 

65.  — A  diameter  of  an  ellipsis  is  any  right  line  passing 
through  the  centre,  and  terminated  by  the  curve.  ( k  l ,  or  m  n , 
Fig.  24.) 

66.  — A  diameter  is  conjugate  to  another  when  it  is  parallel  to 
a  tangent  drawn  at  the  extremity  of  that  other — thus,  the  diame¬ 
ter,  m  n,  (Fig.  24,)  being  parallel  to  the  tangent,  o  p,  is  therefore 
conjugate  to  the  diameter,  k  l. 

67.  — A  double  ordinate  is  any  right  line,  crossing  a  diameter 
of  an  ellipsis,  and  drawn  parallel  to  a  tangent  at  the  extremity  of 
that  diameter,  (i  t ,  Fig.  24.) 

68.  — A  cylinder  is  a  solid  generated  by  the  revolution  of  a 
right-angled  parallelogram,  or  rectangle,  about  one  of  its  sides ; 
and  consequently  the  ends  of  the  cylinder  are  equal  circles. 
( Fig .  25.) 


PRACTICAL  GEOMETRY. 


15 


9 


Fig.  25.  Fig.  26. 


69.  — The  axis  of  a  cylinder  is  a  right  line  passing  through  it, 
from  the  centres  of  the  two  circles  which  form  the  ends. 

70.  — A  segment  of  a  cylinder  is  comprehended  under  three 
planes,  and  the  curved  surface  of  the  cylinder.  Two  of  these 
are  segments  of  circles  :  the  other  plane  is  a  parallelogram,  called 
by  way  of  distinction,  the  plane  of  the  segment.  The  circular 
segments  are  called,  the  ends  of  the  cylinder.  {Fig.  26.) 


PROBLEMS 


RIGHT  LINES  AND  ANGLES. 


71. —  To  bisect  a  line.  Upon  the  ends  of  the  line,  a  b,  {Fig. 
27,)  as  centres,  with  any  distance  for  radius  greater  than  half 


C 


Fig.  27. 


a  b ,  describe  arcs  cutting  each  other  in  c  and  d  ;  draw  the  line, 
c  d ,  and  the  point,  e,  where  it  cuts  a  b,  will  be  the  middle  of  the 
line,  a  b. 

In  practice,  a  line  is  generally  divided  with  the  compasses,  or 
dividers  ;  but  this  problem  is  useful  where  it  is  desired  to  draw, 
at  the  middle  of  another  line,  one  at  right  angles  to  it.  (See 
Art.  85.) 


d 


72. — To  erect  a  perpendicular.  From  the  point,  a ,  {Fig.  28,) 


PRACTICAL  GEOMETRY. 


17 


set  off  any  distance,  as  a  6,  and  the  same  distance  from  a  to  c  ; 
upon  c,  as  a  centre,  with  any  distance  for  radius  greater  than  c  a, 
describe  an  arc  at  d ;  upon  b,  with  the  same  radius,  describe 
another  at  d ;  join  d  and  a,  and  the  line,  d  a ,  will  be  the  per¬ 
pendicular  required. 

This,  and  the  three  following  problems,  are  more  easily  per¬ 
formed  by  the  use  of  the  set-square — (see  Art.  12.)  Yet  they 
are  useful  when  the  operation  is  so  large  that  a  set-square  cannot 
be  used. 


73. —  To  let  fall  a  perpendicular .  Let  a,  [Fig.  29,)  be  the 
point,  above  the  line,  b  c,  from  which  the  perpendicular  is  re¬ 
quired  to  fall.  Upon  a,  with  any  radius  greater  than  a  d ,  de¬ 
scribe  an  arc,  cutting  be  at  e  and  f ;  upon  the  points,  e  and  f 
with  any  radius  greater  than  e  d,  describe  arcs,  cutting  each 
other  at  g  ;  join  a  and  g,  and  the  line,  a  d ,  will  be  the  perpen¬ 
dicular  required. 


74. —  To  erect  a  perpendicular  at  the  end  of  a  line.  Let 
[Fig.  30,)  at  the  end  of  the  line,  c  a,  be  the  point  at  which  the 
perpendicular  is  to  be  erected.  Take  any  point,  as  b,  above  the 

3 


18  • 


AMERICAN  HOUSE-CARPENTER. 


line,  c  a ,  and  with  the  radius,  b  a,  describe  the  arc,  d  a  e; 
through  d  and  b,  draw  the  line,  d  e  ;  join  e  and  a,  then  e  a  will 
be  the  perpendicular  required. 

The  principle  here  made  use  of,  is  a  very  important  one ;  and 
is  applied  in  many  other  cases — (see  Art.  81,  b,  and  Art.  84. 
For  proof  of  its  correctness,  see  Art.  156.) 


Fig.  31. 


74,  a. — A  second  method.  Let  b,  {Fig.  31 ,)  at  the  end  of  the 
line,  a  b,  be  the  point  at  which  it  is  required  to  erect  a  perpen¬ 
dicular.  Upon  b,  with  any  radius  less  than  b  a,  describe  the  arc, 
c  e  d  ;  upon  c,  with  the  same  radius,  describe  the  small  arc  at  e, 
and  upon  e,  another  at  d  ;  upon  e  and  d,  with  the  same  or  any 
other  radius  greater  than  half  e  d,  describe  arcs  intersecting  at  /; 
join  /and  b,  and  the  line,  /  b,  will  be  the  perpendicular  required. 


Fig.  32. 


74,  b. — A  third  method.  Let  b,  {Fig.  32,)  be  the  given  point 
at  which  it  is  required  to  erect  a  perpendicular.  Upon  b,  with  any 
radius  less  than  b  a,  describe  the  quadrant,  d  ef;  upon  d,  with 
the  same  radius,  describe  an  arc  at  e,  and  upon  e,  another  at  c  ; 


PRACTICAL  GEOMETRY. 


19 


through  d  and  e,  draw  d  c,  cutting  the  arc  in  c  ;  join  c  and  b, 
then  c  b  will  be  the  perpendicular  required. 

This  problem  can  be  solved  by  the  six,  eight  and  ten  rule, 
as  it  is  called ;  which  is  founded  upon  the  same  principle  as 
the  problems  at  Art.  103,  104 ;  and  is  applied  as  follows.  Let 
a  d,  {Fig.  30,)  equal  eight,  and  a  e,  six  ;  then,  if  d  e  equals  ten, 
the  angle,  e  a  d,  is  a  right  angle.  Because  the  square  of  six 
and  that  of  eight,  added  together,  equal  the  square  of  ten,  thus  : 
6  x  6  —  36,  and  8  x  8  =  64 ;  36  -f  64  =  100,  and  10  x  10  = 
100.  Any  sizes,  taken  in  the  same  proportion,  as  six,  eight  and 
ten,  will  produce  the  same  effect :  as  3,  4  and  5,  or  12,  16  and 
20.  (See  note  to  Art.  103.) 

By  the  process  shown  at  Fig.  30,  the  end  of  a  board  may  be 
squared  without  a  carpenters’-square.  All  that  is  necessary  is  a 
pair  of  compasses  and  a  ruler.  Let  c  a  be  the  edge  of  the  board, 
and  a  the  point  at  which  it  is  required  to  be  squared.  Take  the 
point,  b ,  as  near  as  possible  at  an  angle  of  forty-five  degrees,  or  on 
amitre-\\ne,  from  a,  and  at  about  the  middle  of  the  board.  This 
is  not  necessary  to  the  working  of  the  problem,  nor  does  it  affect 
its  accuracy,  but  the  result  is  more  easily  obtained.  Stretch  the 
compasses  from  b  to  a ,  and  then  bring  the  leg  at  a  around  to  d  ; 
draw  a  line  from  d,  through  b,  out  indefinitely ;  take  the  dis¬ 
tance,  d  b,  and  place  it  from  b  to  e  ;  join  e  and  a  ;  then  e  a  will 
be  at  right  angles  to  c  a.  In  squaring  the  foundation  of  a  build¬ 
ing,  or  laying-out  a  garden,  a  rod  and  chalk-line  may  be  used 
instead  of  compasses  and  ruler. 

75. —  To  let  fall  a  perpendicular  near  the  end  of  a  line. 
Let  e,  {Fig.  30,)  be  the  point  above  the  line,  c  a,  from  which  the 
perpendicular  is  required  to  fall.  From  e ,  draw  any  line,  as  e  d, 
obliquely  to  the  line,  c  a  ;  bisect  e  d  at  b  ;  upon  b,  with  the 
radius,  b  e,  describe  the  arc,  e  a  d  ;  join  e  and  a  ;  then  e  a  will 
be  the  perpendicular  required. 


76. —  To  make  an  angle ,  (as  e  df  Fig.  33,)  equal  to  a  given 
angle,  (as  b  a  c.)  From  the  angular  point,  a,  with  any  radius, 
describe  the  arc,  be;  and  with  the  same  radius,  on  the  line,  d  e, 


20 


AMERICAN  HOUSE-CARPENTER. 


and  from  the  point,  d,  describe  the  arc,  f  g  ;  take  the  distance, 
b  c,  and  upon  g,  describe  the  small  arc  at  f ;  join  f  and  d  ;  and 
the  angle,  e  df,  will  be  equal  to  the  angle,  b  a  c. 

If  the  given  line  upon  which  the  angle  is  to  be  made,  is  situa¬ 
ted  parallel  to  the  similar  line  of  the  given  angle,  this  may  be 
performed  more  readily  with  the  set-square.  (See  Art.  11.) 


a 


77. —  To  bisect  ari  angle.  Let  a  b  c,  (Fig.  34,)  be  the  angle 
to  be  bisected.  Upon  b,  with  any  radius,  describe  the  arc,  a  c  ; 
upon  a  and  c,  with  a  radius  greater  than  half  a  c,  describe  arcs 
cutting  each  other  at  d  ;  join  b  and  d  ;  and  b  d  will  bisect  the 
angle,  a  b  c,  as  was  required. 

This  problem  is  frequently  made  use  of  in  solving  other  pro¬ 
blems  ;  it  should  therefore  be  well  impressed  upon  the  memory. 


78. —  To  trisect  a  right  angle.  Upon  a,  (Fig.  35,)  with  any 
radius,  describe  the  arc,  b  c  ;  upon  b  and  c,  with  the  same  radius, 
describe  arcs  cutting  the  arc,  b  c,  at  d  and  e  ;  from  d  and  e,  draw 
lines  to  a,  and  they  will  trisect  the  angle  as  was  required. 

The  truth  of  this  is  made  evident  by  the  following  operation. 
Divide  a  circle  into  quadrants  :  also,  take  the  radius  in  the  divi¬ 
ders,  and  space  off  the  circumference.  This  will  divide  the 
circumference  into  just  six  parts.  A  semi-circumference,  there- 


PRACTICAL  GEOMETRY. 


21 


fore,  is  equal  to  three,  and  a  quadrant  to  one  and  a  half  of  those 
parts.  The  radius,  therefore,  is  equal  to  f  of  a  quadrant ;  and 
this  is  equal  to  a  right  angle. 


b  d  c 


Fig.  36. 

79. —  Through  a  given  point ,  to  draw  a  line  parallel  to  a 
given  line.  Let  a,  (Fig.  36,)  be  the  given  point,  and  b  c  the 
given  line.  Upon  any  point,  as  d ,  in  the  line,  b  c,  with  the 
radius,  d  a,  describe  the  arc,  a  c;  upon  a,  with  the  same  radius, 
describe  the  arc,  d  e  ;  make  d  e  equal  to  a  c  ;  through  e  and  a, 
draw  the  line,  e  a  ;  which  will  be  the  line  required. 

This  is  upon  the  same  principle  as  Art.  76. 


80. —  To  divide  a  given  line  into  any  number  of  equal  parts. 

Let  a  Z>,  (Fig.  37,)  be  the  given  line,  and  5  the  number  of  parts. 

Draw  a  c,  at  any  angle  to  a  b  ;  on  a  c,  from  a,  set  off  5  equal 

parts  of  any  length,  as  at  1,  2,  3,  4  and  c  ;  join  c  and  b  ;  through 

the  points,  1,  2,  3  and  4,  draw  1  e,  2  /,  3  g  and  4  A,  parallel  to 

c  b  ;  which  will  divide  the  line,  a  b ,  as  was  required. 

The  lines,  a  b  and  a  c,  are  divided  in  the  same  proportion. 
(See  Art.  109.) 


THE  CIRCLE. 

81. — To  find  the  centre  of  a  circle.  Draw  any  chord,  as  a  b , 


22 


AMERICAN  HOUSE-CARPENTER. 


c 


{Fig.  38,)  and  bisect  it  with  the  perpendicular,  c  d  ;  bisect  c  d 
with  the  line,  e  /,  as  at  g  ;  then  g  is  the  centre  as  was  required. 


e 


81,  a. — A  second  method.  Upon  any  two  points  in  the  cir¬ 
cumference  nearly  opposite,  as  a  and  b,  {Fig.  39,)  describe  arcs 
cutting  each  other  at  c  and  d ;  take  any  other  two  points,  as  e 
and  and  describe  arcs  intersecting  as  at  g  and  h  ;  join  g  and  h, 
and  c  and  d  ;  the  intersection,  o,  is  the  centre. 

This  is  upon  the  same  principle  as  Art.  85. 


81,  b. — A  third  method.  Draw  any  chord,  as  a  b}  {Fig.  40,) 


PRACTICAL  GEOMETRY. 


23 


and  from  the  point,  a ,  draw  a  c,  at  right  angles  to  a  b  ;  join 
c  and  b  ;  bisect  c  b  at  d — which  will  be  the  centre  of  the  circle. 

If  a  circle  be  not  too  large  for  the  purpose,  its  centre  may  very 
readily  be  ascertained  by  the  help  of  a  carpenters’ -square,  thus  : 
app'  y  the  corner  of  the  square  to  any  point  in  the  circumference, 
as  at  a ;  by  the  edges  of  the  square,  (which  the  lines,  a  b  and 
a  c,  represent,)  draw  lines  cutting  the  circle,  as  at  b  and  c  ;  join 
b  and  c  ;  then  if  b  c  is  bisected,  as  at  d,  the  point,  d,  will  be  the 
centre.  (See  Art.  156.) 


C 


82. — At  a  given  'point  in  a  circle ,  to  draw  a  tangent  thereto. 
Let  a,  {Fig.  41,)  be  the  given  point,  and  b  the  centre  of  the  cir¬ 
cle.  Join  a  and  b  ;  through  the  point,  a,  and  at  right  angles  to 
a  b,  draw  c  d  ;  c  d  is  the  tangent  required. 


Fig.  42. 


83.  —  The  same ,  without  making  use  of  the  centre  of  the 
circle.  Let  a,  {Fig.  42,)  be  the  given  point.  From  a,  set  off 
any  distance  to  b ,  and  the  same  from  b  to  c ;  join  a  and  c  ; 
upon  a,  with  a  b  for  radius,  describe  the  arc,  d  b  e  ;  make  d  b 
equal  to  be;  through  a  and  d,  draw  a  line  ;  this  will  be  the 
tangent  required. 

84.  — A  circle  and  a  tangent  given,  to  find  the  point  of  con¬ 
tact.  From  any  point,  as  a ,  {Fig.  43,)  in  the  tangent,  b  c,  draw 


24 


AMERICAN  HOUSE-CARPENTER. 


a  line  to  the  centre  d  ;  bisect  a  d  at  e  ;  upon  e,  with  the  radius, 

e  a,  describe  the  arc,  a  f  d  ;  f  is  the  point  of  contact  required. 

If  /  and  d  were  joined,  the  line  would  form  right  angles  with 
the  tangent,  b  c.  (See  Art.  156.) 


b 


85. —  Through  any  three  points  not  in  a  straight  line ,  to 
draw  a  circle.  Let  a,  b  and  c,  {Fig.  44,)  be  the  three  given 
points.  Upon  a  and  b,  with  any  radius  greater  than  half  a  b , 
describe  arcs  intersecting  at  d  and  e  ;  upon  b  and  c,  with  any 
radius  greater  than  half  b  c,  describe  arcs  intersecting  at  f  and  g  ; 
through  d  and  e,  draw  a  right  line,  also  another  through  /andg-; 
upon  the  intersection,  h,  with  the  radius,  h  a,  describe  the  circle, 
a  b  c,  and  it  will  be  the  one  required. 


Fig.  45. 


PRACTICAL  GEOMETRY. 


25 


86. —  Three  -points  not  in  a  straight  line  being  given ,  to  find 
a  fourth  that  shall ,  with  the  three ,  lie  in  the  circumference  of 
a  circle.  Let  a  b  c,  {Fig.  45,)  be  the  given  points.  Connect 
them  with  right  lines,  forming  the  triangle,  a  c  b  ;  bisect  the 
angle,  c  b  a,  {Art.  77,)  with  the  line,  b  d  ;  also  bisect  c  a  in  e, 
and  erect  e  d ,  perpendicular  to  a  c,  cutting  b  d  in  d  ;  then  d  is 
the  fourth  point  required. 

A  fifth  point  may  be  found,  as  at/,  by  assuming  a,  d  and  b, 
as  the  three  given  points,  and  proceeding  as  before.  So,  also, 
any  number  of  points  may  be  found  ;  simply  by  using  any  three 
already  found.  This  problem  will  be  serviceable  in  obtaining 
short  pieces  of  very  flat  sweeps.  (See  Art.  311.) 


C 


87. —  To  describe  a  segment  of  a  circle  by  a  set-triangle. 
Let  a  b,  {Fig.  46,)  be  the  chord,  and  c  d  the  height  of  the  seg¬ 
ment.  Secure  two  straight-edges,  or  rulers,  in  the  position,  c  e 
and  c  f  by  nailing  them  together  at  c,  and  affixing  a  brace  from 
e  to  f ;  put  in  pins  at  a  and  b  ;  move  the  angular  point,  c,  in 
the  direction,  a  c  b  ;  keeping  the  edges  of  the  triangle  hard 
against  the  pins,  a  and  b ;  a  pencil  held  at  c  will  describe  the 
arc,  a  c  b. 

If  the  angle  formed  by  the  rulers  at  c  be  a  right  angle,  the 
segment  described  will  be  a  semi-circle.  This  problem  is  useful 
in  describing  centres  for  brick  arches,  when  they  are  required  to 
be  rather  flat.  Also,  for  the  head  hanging-style  of  a  window- 
frame,  where  a  brick  arch,  instead  of  a  stone  lintel,  is  to  be 
placed  over  it. 


eg  1  2  3  c  3  2  1  h  J 


Fig.  47. 

4 


26 


AMERICAN  HOUSE-CARPENTER. 


88. —  To  describe  the  segment  of  a  circle  by  intersection  of 
lines.  Let  a  b,  {Fig.  47,)  be  the  chord,  and  c  d  the  height  of 
the  segment.  Through  c,  draw  e  f  parallel  to  a  b  ;  draw  b  f  at 
right  angles  to  c  b  ;  make  c  e  equal  to  c  f;  draw  a  g  and  b  h , 
at  right  angles  to  a  b  ;  divide  c  e,  c  f  d  a,  d  b,  a  g  and  b  h ,  each 
into  a  like  number  of  equal  parts,  as  four ;  draw  the  lines,  1  1, 
2  2,  &c.,  and  from  the  points,  o,  o  and  o,  draw  lines  to  c  ;  at  the 
intersection  of  these  lines,  trace  the  curve,  a  c  b,  which  will  be 
the  segment  required. 

In  very  large  work,  or  in  laying  out  ornamented  gardens,  &c., 
this  will  be  found  useful ;  and  where  the  centre  of  the  proposed 
arc  of  a  circle  is  inaccessible,  it  will  be  invaluable.  (To  trace 
the  curve,  see  note  at  Art.  117.) 


b 


89. — In  a  given  angle ,  to  describe  a  tanged  curve.  Let  a 
b  c,  {Fig.  48,)  be  the  given  angle,  and  1  in  the  line,  a  b,  and  5 
in  the  line,  b  c,  the  termination  of  the  curve.  Divide  1  b  and  b  5 
into  a  like  number  of  equal  parts,  as  at  1,  2,  3,  4  and  5  ;  join  1 
and  1,  2  and  2,  3  and  3,  &c.  ;  and  a  regular  curve  will  be  formed 
that  will  be  tangical  to  the  line,  a  b,  at  the  point,  1,  and  to  b  c 
at  5. 

This  is  of  much  use  in  stair-building,  in  easing  the  angles 
formed  between  the  wall-string  and  base  of  the  hall,  also  between 
the  front  string  and  level  facia,  and  in  many  other  instances. 
The  curve  is  not  circular,  but  of  the  form  of  the  parabola,  {Fig. 
93 ;)  yet  in  large  angles  the  difference  is  not  perceptible.  This 
problem  can  be  applied  to  describing  segments  of  circles  for  door- 


e 


a  c  b 

Fig.  49. 


PRACTICAL  GEOMETRY. 


27 


heads,  window-heads,  &c.,  to  rather  better  advantage  than  Art. 
87.  For  instance,  let  a  b ,  ( Fig .  49,)  be  the  width  of  the  open¬ 
ing,  and  c  d  the  height  of  the  arc.  Extend  c  d,  and  make  d  e 
equal  to  c  d  ;  join  a  and  e,  also  e  and  b  ;  and  proceed  as  direct¬ 
ed  at  Art.  89. 


Fig.  50. 


90. —  To  describe  a  circle  within  any  given  triangle ,  so  that 
the  sides  of  the  triangle  shall  be  tangical.  Let  a  b  c,  (Fig. 
50,)  be  the  given  triangle.  Bisect  the  angles,  a  and  b ,  according 
to  Art.  77 ;  upon  d,  the  point  of  intersection  of  the  bisecting 
lines,  with  the  radius,  d  e,  describe  the  required  circle. 


h 


91. — About  a  given  circle,  to  describe  an  equi-lateral  tri¬ 
angle.  Let  a  db  c,  (Fig.  51,)  be  the  given  circle.  Draw  the 
diameter,  c  d ;  upon  d,  with  the  radius  of  the  given  circle,  de¬ 
scribe  the  arc,  a  e  b ;  join  a  and  b  ;  draw  f  g,  at  right  angles  to 
d  c  ;  make  f  c  and  c  g,  each  equal  to  a  b  ;  from/,  through  a, 
draw  /  h,  also  from  g,  through  b,  draw  g  h  ;  then  f  g  h  will  be 
the  triangle  required. 


28 


AMERICAN  HOUSE-CARPENTER. 


e 


92. —  To  find  a  right  line  nearly  equal  to  the  circumference 
of  a  circle.  Let  abed ,  {Fig.  52,)  be  the  given  circle.  Draw 
the  diameter,  a  c  ;  on  this  erect  an  equi-lateral  triangle,  a  e  c, 
according  to  Art.  96  ;  draw  g  f  parallel  to  a  c  ;  extend  e  c  to  f 
also  e  a  to  g ;  then  gf  will  be  nearly  the  length  of  the  semi¬ 
circle,  ad  c  ;  and  twice  g  f  will  nearly  equal  the  circumference 
of  the  circle,  abed,  as  was  required. 

Lines  drawn  from  e,  through  any  points  in  the  circle,  as  o,  o 
and  o,  to  p,  p  and  p,  will  divide  g  f  in  the  same  way  as  the  semi¬ 
circle,  a  d  c,  is  divided.  So,  any  portion  of  a  circle  may  be 
transferred  to  a  straight  line.  This  is  a  very  useful  problem, 
and  should  be  well  studied ;  as  it  is  frequently  used  to  solve 
problems  on  stairs,  domes,  & c. 


b 


Fig.  53. 


92,  a. — Another  method.  Let  a  bf  c,  {Fig.  53,)  be  the  given 
circle.  Draw  the  diameter,  a  c  ;  from  d,  the  centre,  and  at  right 
angles  to  a  c,  draw  d  b ;  join  b  and  c  ;  bisect  be  at  e;  from  d, 
through  e,  draw  df;  then  e  /  added  to  three  times  the  diameter, 


PRACTICAL  GEOMETRY. 


29 


will  equal  the  circumference  of  the  circle  within  the  4T)Vir  part  of 
its  length. 

POLYGONS,  &C. 

93. —  Within  a  given  circle ,  to  inscribe  an  equi-lateral  tri- 
angle)  hexagon  or  dodecagon.  Let  abed ,  (Fig.  54,)  be  the 


b 


d 

Fig.  54. 


given  circle.  Draw  the  diameter,  b  d  ;  upon  b,  with  the  radius 
of  the  given  circle,  describe  the  arc,  a  e  c  ;  join  a  and  c,  also  a 
and  d ,  and  c  and  d — and  the  triangle  is  completed.  For  the 
hexagon :  from  a,  also  from  c,  through  e,  draw  the  lines,  a  f 
and  eg;  join  a  and  6,  b  and  c,  c  and/,  &c.,  and  the  hexagon  is 
completed.  The  dodecagon  may  be  formed  by  bisecting  the 
sides  of  the  hexagon. 

Each  side  of  a  regular  hexagon  is  exactly  equal  to  the  radius 
of  the  circle  that  circumscribes  the  figure.  For  the  radius  is 
equal  to  a  chord  of  an  arc  of  60  degrees  ;  and,  as  every  circle  is 
supposed  to  be  divided  into  360  degrees,  there  is  just  6  times  60, 
or  6  arcs  of  60  degrees,  in  the  whole  circumference.  A  line 
drawn  from  each  angle  of  the  hexagon  to  the  centre,  (as  in  the 
figure,)  divides  it  into  six  equal,  equi-lateral  triangles. 


2 


3  ft 


c  7 


6  d 


Fig.  55. 


30 


AMERICAN  HOUSE-CARPENTER. 


94. —  Within  a  square  to  inscribe  an  octagon.  Let  abed, 
{Fig.  55,)  be  the  given  square.  Draw  the  diagonals,  a  d  and 
be;  upon  a,  b,  c  and  d,  with  a  e  for  radius,  describe  arcs  cut¬ 
ting  the  sides  of  the  square  at  1,  2,  3,  4,  5,  6,  7  and  8  ;  join  1 
and  2,  3  and  4,  5  and  6,  &c.,  and  the  figure  is  completed. 

In  order  to  eight-square  a  hand-rail,  or  any  piece  that  is  to  be 
afterwards  rounded,  draw  the  diagonals,  a  d  and  b  c,  upon  the 
end  of  it,  after  it  has  been  squared-up.  Set  a  gauge  to  the  dis¬ 
tance,  a  e,  and  run  it  upon  the  whole  length  of  the  stuff,  from 
each  corner  both  ways.  This  will  show  how  much  is  to  be 
chamfered  off,  in  order  to  make  the  piece  octagonal. 

e  e  e 


95.  —  Within  a  given  circle  to  inscribe  any  regular  polygon. 
Let  a  b  c  2,  {Fig.  56,  57  and  58,)  be  given  circles.  Draw  the 
diameter,  a  c  ;  upon  this,  erect  an  equi-lateral  triangle,  a  e  c, 
according  to  Art.  96 ;  divide  a  c  into  as  many  equal  parts  as  the 
polygon  is  to  have  sides,  as  at  1,  2,  3,  4,  &c.  ;  from  e ,  through 
each  even  number,  as  2,  4,  6,  &c.,  draw  lines  cutting  the  circle 
in  the  points,  2,  4,  &c. ;  from  these  points  and  at  right  angles  to 
a  c,  draw  lines  to  the  opposite  part  of  the  circle  ;  this  will  give 
the  remaining  points  for  the  polygon,  as  b,  /,  &c. 

In  forming  a  hexagon,  the  sides  of  the  triangle  erected  upon 
a  c,  (as  at  Fig.  57,)  mark  the  points,  b  and/. 

96.  —  Upon  a  given  line  to  construct  an  equi-lateral  triangle. 
Let  a  b,  {Fig.  59,)  be  the  given  line.  Upon  a  and  b,  with  a  b 


PRACTICAL  GEOMETRY. 


31 


c 


A 


a 


b 


Fig.  59. 


for  radius,  describe  arcs  intersecting  at  c ;  join  a  and  c,  also  c 
and  b  ;  then  a  cb  will  be  the  triangle  required. 


b 


a 


Fig.  60 


97. —  To  describe  an  equi-lateral  rectangle ,  or  square.  Let 
a  b ,  {Fig.  60,)  be  the  length  of  a  side  of  the  proposed  square. 
Upon  a  and  6,  with  a  b  for  radius,  describe  the  arcs,  a  d  and  be; 
bisect  the  arc,  a  e,  in  /;  upon  e,  with  e /for  radius,  describe  the 
arc,  c  f  d ;  join  a  and  c,  c  and  d ,  d  and  b  ;  then  a  c  db  will 
be  the  square  required. 


Fig-  61.  Fig.  62.  Fig.  63. 


98. —  Upon  a  given  line  to  describe  any  regular  polygon. 
Let  a  b,  {Fig.  61,  62  and  63,)  be  given  lines,  equal  to  a  side  of 
the  required  figure.  From  b,  draw  b  c,  at  right  angles  to  a  b  ; 
upon  a  and  b,  with  a  b  for  radius,  describe  the  arcs,  a  c  d  and 


32 


AMERICAN  HOUSE-CARPENTER. 


/  eb  ;  divide  a  c  into  as  many  equal  parts  as  the  polygon  is  to 
have  sides,  and  extend  those  divisions  from  c  towards  d  ;  from 
the  second  point  of  division  counting  from  c  towards  a,  as  3, 
{Fig-.  61,)  4,  {Fig.  62,)  and  5,  {Fig.  63,)  draw  a  line  to  b  ;  take 
the  distance  from  said  point  of  division  to  a ,  and  set  it  from  b 
to  e  ;  join  e  and  a  ;  upon  the  intersection,  o,  with  the  radius, 
o  a ,  describe  the  circle,  a  f  d  b  ;  then  radiating  lines,  drawn 
from  b  through  the  even  numbers  on  the  arc,  a  d,  will  cut  the 
circle  at  the  several  angles  of  the  required  figure. 

In  the  hexagon,  {Fig.  62,)  the  divisions  on  the  arc,  a  d,  are 
not  necessary  ;  for  the  point,  o,  is  at  the  intersection  of  the  arcs, 
a  d  and  /  6,  the  points,  /  and  d,  are  determined  by  the  intersec¬ 
tion  of  those  arcs  with  the  circle,  and  the  points  above,  g  and  k, 
can  be  found  by  drawing  lines  from  a  and  b ,  through  the  centre, 
o.  In  polygons  of  a  greater  number  of  sides  than  the  hexagon, 
the  intersection,  o,  comes  above  the  arcs  ;  in  such  case,  therefore, 
the  lines,  a  e  and  b  5,  {Fig.  63,)  have  to  be  extended  before  they 
will  intersect. 


Fig.  64. 

99. —  To  construct  a  triangle  whose  sides  shall  be  severally 
equal  to  three  given  lines.  Let  a.  b  and  c,  {Fig.  64,)  be  the 
given  lines.  Draw  the  line,  d  e,  and  make  it  equal  to  c  ;  upon 
e,  with  b  for  radius,  describe  an  arc  at/;  upon  d,  with  a  for 
radius,  describe  an  arc  intersecting  the  other  at/;  join  d  and/, 
also  /  and  e  ;  then  dfe  will  be  the  triangle  required. 


PRACTICAL  GEOMETRY. 


33 


100. —  To  construct  a  figure  equal  to  a  given ,  right-lined 
figure.  Let  abed,  (Fig.  65,)  be  the  given  figure.  Make  e  f, 
(Fig.  66,)  equal  to  c  d  ;  upon  /  with  d  a  for  radius,  describe  an 
arc  at  g ;  upon  e,  with  c  a  for  radius,  describe  an  arc  intersecting 
the  other  at  g  ;  join  g  and  e  ;  upon  /  and  g,  with  d  b  and  a  b 
for  radius,  describe  arcs  intersecting  at  h  ;  join  g  and  h ,  also  h 
and/;  then  Fig.  66  will  every  way  equal  Fig.  65. 

So,  right-lined  figures  of  any  number  of  sides  may  be  copied, 
by  first  dividing  them  into  triangles,  and  then  proceeding  as 
above.  The  shape  of  the  floor  of  any  room,  or  of  any  piece  of 
land,  &c.,  may  be  accurately  laid  out  by  this  problem,  at  a  scale 
upon  paper ;  and  the  contents  in  square  feet  be  ascertained  by 
the  next. 


a 


101. —  To  make  a  'parallelogram  equal  to  a  given  triangle. 
Let  a  b  c,  (Fig.  67,)  be  the  given  triangle.  From  a,  draw  a  d, 
at  right  angles  to  be;  bisect  ad  in  e  ;  through  e,  draw  /  g, 
parallel  to  be;  from  b  and  c,  draw  b  f  and  c  g,  parallel  to  d  e  ; 
then  bfgc  will  be  a  parallelogram  containing  a  surface  exactly 
equal  to  that  of  the  triangle,  a  b  c. 

Unless  the  parallelogram  is  required  to  be  a  rectangle,  the  lines, 
b  f  and  c  g,  need  not  be  drawn  parallel  to  d  e.  If  a  rhomboid  is 
desired,  they  may  be  drawn  at  an  oblique  angle,  provided  they 
be  parallel  to  one  another.  To  ascertain  the  area  of  a  triangle, 
multiply  the  base,  b  c ,  by  half  the  perpendicular  height,  d  a.  In 
doing  this,  it  matters  not  which  side  is  taken  for  base. 


34 


AMERICAN  HOUSE-CARPENTER. 


102. — A  'parallelogram  being  given ,  to  construct  another 
equal  to  it,  and  having  a  side  equal  to  a  given  line.  Let  A, 
{Fig.  68,)  be  the  given  parallelogram,  and  B  the  given  line. 
Produce  the  sides  of  the  parallelogram,  as  at  a,  b,  c  and  d  ;  make 
e  d  equal  to  B  ;  through  d,  draw  c  /,  parallel  to  g  b  ;  through 
c,  draw  the  diagonal,  c  a  ;  from  a ,  draw  a  f,  parallel  to  e  d ; 
then  C  will  be  equal  to  A.  (See  Art.  144.) 


103. —  To  make  a  square  equal  to  two  or  more  given  squares. 
Let  A  and  B,  {Fig.  69,)  be  two  given  squares.  Place  them  so 
as  to  form  a  right  angle,  as  at  a  ;  join  b  and  c  ;  then  the  square, 
C,  formed  upon  the  line,  b  c ,  will  be  equal  in  extent  to  the  squares, 
A  and  B,  added  together.  Again  :  if  a  b,  {Fig.  70,)  be  equal  to 


Fig.  70. 


the  side  of  a  given  square,  c  a,  placed  at  right  angles  to  a  b ,  be  the 
side  of  another  given  square,  and  c  d ,  placed  at  right  angles  to 


PRACTICAL  GEOMETRY. 


35 


c  6,  be  the  side  of  a  third  given  square  ;  then  the  square,  A, 
formed  upon  the  line,  d  b,  will  be  equal  to  the  three  given 
squares.  (See  Art.  157.) 

The  usefulness  and  importance  of  this  problem  are  proverbial. 
To  ascertain  the  length  of  braces  and  of  rafters  in  framing,  the 
length  of  stair-strings,  &c.,  are  some  of  the  purposes  to  which  it 
may  be  applied  in  carpentry.  (See  note  to  Art.  74,  b.)  If  the 
length  of  any  two  sides  of  a  right-angled  triangle  is  known,  that 
of  the  third  can  be  ascertained.  Because  the  square  of  the 
hypothenuse  is  equal  to  the  united  squares  of  the  two  sides  that 
contain  the  right  angle. 

(1.) — The  two  sides  containing  the  right  angle  being  known, 
to  find  the  hypothenuse.  Rule. — Square  each  given  side,  add 
the  squares  together,  and  from  the  product  extract  the  square- 
root  :  this  will  be  the  answer.  For  instance,  suppose  it  were 
required  to  find  the  length  of  a  rafter  for  a  house,  34  feet  wide, — 
the  ridge  of  the  roof  to  be  9  feet  high,  above  the  level  of  the 
wall-plates.  Then  17  feet,  half  of  the  span,  is  one,  and  9  feet, 
the  height,  is  the  other  of  the  sides  that  contain  the  right  angle. 
Proceed  as  directed  by  the  rule : 

17  9 

17  9 


119  81  =  square  of  9. 

17  289  =  square  of  17. 


289  =■  square  of  17.  370  Product. 

1  )  370  (  19‘235  +  =  square-root  of  370  ;  equal  19  feet,  2\  in. 
1  1  nearly :  which  would  be  the  required 

—  -  length  of  the  rafter. 

29  )  270 
9  261 


382)- -900 
2  764 


3843 )  13600 
3  11529 


38465)-  207100 
192325 


(By  reference  to  the  table  of  square-roots  in  the  appendix,  the 
root  ot  almost  any  number  may  be  found  ready  calculated.) 


36 


AMERICAN  HOUSE-CARPENTER. 


Again :  suppose  it  be  required,  in  a  frame  building,  to  find  the 
length  of  a  brace,  having  a  run  of  three  feet  each  way  from  the 
point  of  the  right  angle.  The  length  of  the  sides  containing  the 
right  angle  will  be  each  3  feet :  then,  as  before — 

3 

3 

9  =  square  of  one  side. 

3  times  3  =  9  =  square  of  the  other  side. 

1 8  Product :  the  square-root  of  which  is  4'2426  -{-  ft., 
or  4  feet,  2  inches  and  fths.  full.  • 

(2.) — The  hypothenuse  and  one  side  being  known,  to  find  the 
other  side.  Rule. — Subtract  the  square  of  the  given  side  from 
the  square  of  the  hypothenuse,  and  the  square-root  of  the  product 
will  be  the  answer.  Suppose  it  were  required  to  ascertain  the 
greatest  perpendicular  height  a  roof  of  a  given  span  may  have, 
when  pieces  of  timber  of  a  given  length  are  to  be  used  as  rafters. 
Let  the  span  be  20  feet,  and  the  rafters  of  3  X  4  hemlock  joist. 
These  come  about  13  feet  long.  The  known  hypothenuse, 
then,  is  13  feet,  and  the  known  side,  10  feet — that  being  half  the 
span  of  the  building. 

13 

13 


39 

13 


169  =  square  of  hypothenuse. 

10 'times  10  =  100  =  square  of  the  given  side. 

69  Product :  the  square-root  of  which  is  8 
•3066  +  feet,  or  8  feet,  3  inches  and  ^ths.  full.  This  will  be 
the  greatest  perpendicular  height,  as  required.  Again  :  suppose 
that  in  a  story  of  8  feet,  from  floor  to  floor,  a  step-ladder  is  re¬ 
quired,  the  strings  of  which  are  to  be  of  plank,  12  feet  long  ;  and 
it  is  desirable  to  know  the  greatest  run  such  a  length  of  string 
will  afford.  In  this  case,  the  two  given  sides  are— hypothenuse 
12,  perpendicular  8  feet. 

12  times  12  =  144  ==  square  of  hypothenuse. 

8  times  8  =  64  =  square  of  perpendicular. 

80  Product :  the  square-root  of  which  is  8‘9442-p 
feet,  or  8  feet,  11  inches  and  ^ths. — the  answer,  as  required. 


PRACTICAL  GEOMETRY. 


37 


Many  other  cases  might  he  adduced  to  show  the  utility  of  this 
problem.  A  practical  and  ready  method  of  ascertaining  the 
length  of  braces,  rafters,  &c.,  when  not  of  a  great  length,  is  to 
apply  a  rule  across  the  carpenters’-square.  Suppose,  for  the 
length  of  a  rafter,  the  base  be  12  feet  and  the  height  7.  Apply 
the  rule  diagonally  on  the  square,  so  that  it  touches  12  inches 
from  the  corner  on  one  side,  and  7  inches  from  the  corner  on  the 
other.  The  number  of  inches  on  the  rule,  which  are  intercepted 
by  the  sides  of  the  square,  13|-  nearly,  will  be  the  length  of  the 
rafter  in  feet ;  viz,  13  feet  and  8ths  of  a  foot.  If  the  dimensions 
are  large,  as  30  feet  and  20,  take  the  half  of  each  on  the  sides  of 
the  square,  viz,  15  and  10  inches  ;  then  the  length  in  inches 
across,  will  be  one-half  the  number  of  feet  the  rafter  is  long. 
This  method  is  just  as  accurate  as  the  preceding  ;  but  when 
the  length  of  a  very  long  rafter  is  sought,  it  requires  great  care 
and  precision  to  ascertain  the  fractions.  For  the  least  variation 
on  the  square,  or  in  the  length  taken  on  the  rule,  would  make 
perhaps  several  inches  difference  in  the  length  of  the  rafter. 
For  shorter  dimensions,  however,  the  result  will  be  true  enough. 


Pig.  71. 


104. —  To  ynake  a  circle  equal  to  two  given  circles.  Let  A 
and  B,  {Fig.  71,)  be  the  given  circles.  In  the  right-angled  tri¬ 
angle,  a  b  c,  make  a  b  equal  to  the  diameter  of  the  circle,  B:  and 
c  b  equal  to  the  diameter  of  the  circle,  A  ;  then  the  hypothenuse, 


38 


AMERICAN  HOUSE-CARPENTER. 


a  c,  will  be  the  diameter  of  a  circle,  C,  which  will  be  equal  in 
area  to  the  two  circles,  A  and  B,  added  together. 

Any  polygonal  figure,  as  A,  {Fig.  72,)  formed  on  the  hypo- 
thenuse  of  a  right-angled  triangle,  will  be  equal  to  two  similar 
figures,*  as  B  and  C,  formed  on  the  two  legs  of  the  triangle. 


f  s/ 

\ 

A 

J 

e 

Fig.  73. 


105. —  To  construct  a  square  equal  to  a  given  rectangle. 
Let  A,  (Fig.  73,)  be  the  given  rectangle.  Extend  the  side,  a  b , 
and  make  b  c  equal  to  be;  bisect  a  c  in/,  and  upon/,  with  the 
radius,  /  a,  describe  the  semi-circle,  age;  extend  e  b,  till  it 
cuts  the  curve  in  g  ;  then  a  square,  b  g  h  d,  formed  on  the  line, 
b  g ,  will  be  equal  in  area  to  the  rectangle,  A. 


105,  a. — Another  method.  Let  A,  (Fig.  74,)  be  the  given 
rectangle.  Extend  the  side,  a  b,  and  make  a  d  equal  to  a  c  ; 

*  Similar  figures  are  such  as  have  their  several  angles  respectively  equal,  and  their 
sides  respectively  proportionate. 


PRACTICAL  GEOMETRY. 


39 


#■ 

bisect  a  d  in  e  ;  upon  e,  with  the  radius,  e  a ,  describe  the  semi¬ 
circle,  a  f  d ;  extend  g  b  till  it  cuts  the  curve  in  f ;  join  a  and 
f ;  then  the  square,  B ,  formed  on  the  line,  afi  will  be  equal  in 
area  to  the  rectangle,  A.  (See  Art.  156  and  157.) 

106. —  To  form  a  square  equal  to  a  given  triangle.  Let  a  b , 
{Fig.  73,)  equal  the  base  of  the  given  triangle,  and  b  e  equal 
half  its  perpendicular  height,  (see  Fig.  67 ;)  then  proceed  as 
directed  at  Art.  105. 

A -  - - 1 -■ 

B - 


107. —  Two  right  lines  being  given ,  to  find  a  third  propor¬ 
tional  thereto.  Let  A  and  B ,  {Fig.  75,)  be  the  given  lines. 
Make  a  b  equal  to  A  ;  from  a,  draw  a  c,  at  any  angle  with  a  b  ; 
make  a  c  and  a  d  each  equal  to  B  ;  join  c  and  b  ;  from  d,  draw 
d  e,  parallel  to  c  b  ;  then  a  e  will  be  the  third  proportional  re¬ 
quired.  That  is,  a  e  bears  the  same  proportion  to  B ,  as  B  does 
to  A. 

A---1  ■  ' 

B - - - 

C _ _ 


c 


108. —  Three  right  lines  being  given ,  to  find  a  fourth  pro¬ 
portional  thereto.  Let  A,  B  and  C,  {Fig.  76,)  be  the  given 
lines.  Make  a  b  equal  to  A  ;  from  «,  draw  a  c,  at  any  angle 
with  a  b;  make  a  c  equal  to  B ,  and  a  e  equal  to  C ;  join  c  and 
b  ;  from  e,  draw  e  f  parallel  to  c  b  ;  then  a  f  will  be  the  fourth 
proportional  required.  That  is,  a  f  bears  the  same  proportion 
to  C,  as  B  does  to  A. 


40 


AMERICAN  HOUSE-CARPENTER. 


To  apply  this  problem,  suppose  the  two  axes  of  a  given  ellipsis, 
and  the  longer  axis  of  a  proposed  ellipsis  are  given.  Then,  by 
this  problem,  the  length  of  the  shorter  axis  to  the  proposed  ellip¬ 
sis,  can  be  found  ;  so  that  it  will  bear  the  same  proportion  to  the 
longer  axis,  as  the  shorter  of  the  given  ellipsis  does  to  its  longer. 
(See  also,  Art.  126.) 


109. — A  line  with  certain  divisions  being  given,  to  divide 
another,  longer  or  shorter,  given  line  in  the  same  proportion. 
Let  A,  {Fig.  77,)  be  the  line  to  be  divided,  and  B  the  line  with 
its  divisions.  Make  a  b  equal  to  B,  with  all  its  divisions,  as  at 
1,  2,  3,  &c. ;  from  a,  draw  a  c,  at  any  angle  with  ah  ;  make  a  c 
equal  to  A  ;  join  c  and  b  ;  from  the  points,  1,  2,  3,  &c.,  draw 
lines,  parallel  to  c  b  ;  then  these  will  divide  the  line,  a  c,  in  the 
same  proportion  as  B  is  divided — as  was  required. 

This  problem  will  be  found  useful  in  proportioning  the  mem¬ 
bers  of  a  proposed  cornice,  in  the  same  proportion  as  those  of  a 
given  cornice  of  another  size.  (See  Art.  243  and  244.)  So  of 
a  pilaster,  architrave,  &c. 


110. — Between  two  given  right  lines,  to  find  a  mean  pro¬ 
portional.  Let  A  and  B,  {Fig.  78,)  be  the  given  lines.  On 
the  line,  a  c,  make  a  b  equal  to  A,  and  b  c  equal  to  B  ;  bisect  a 
cine  ;  upon  e,  with  e  a  for  radius,  describe  the  semi-circle,  a  d 


PRACTICAL  GEOMETRY.  41 

c  ;  at  b,  erect  b  d,  at  right  angles  to  a  c;  then  b  d  will  be  the 
mean  proportional  between  A  and  B. 

For  an  application  of  this  problem,  see  Art.  105. 

CONIC  SECTIONS. 

HI* — If  a  cone,  standing  upon  a  base  that  is  at  right  angles 
with  its  axis,  be  cut  by  a  plane,  perpendicular  to  its  base  and 
passing  through  its  axis,  the  section  will  be  an  isoceles  triangle  ; 
(as  a  b  c,  Fig.  79  ;)  and  the  base  will  be  a  semi-circle.  If  a 


cone  be  cut  by  a  plane  in  the  direction,  e  f,  the  section  will  be 
an  ellipsis  ;  if  in  the  direction,  m  l ,  the  section  will  be  a  para¬ 
bola  ;  and  if  in  the  direction,  r  o,  an  hyperbola.  (See  Art.  56 
to  60.)  If  the  cutting  planes  be  at  right  angles  with  the  plane, 
a  b  c,  then — 

112. —  To  find  the  axis  of  the  ellipsis,  bisect  e  /,  (Fig.  79,) 
in  g  ;  through  g,  draw  h  i,  parallel  to  a  b  ;  bisect  h  i  i nj  ;  upon 
j,  with  j  h  for  radius,  describe  the  semi-circle,  h  k  i  ;  from  g, 
draw  g  k,  at  right  angles  to  h  i  ;  then  twice  g  k  will  be  the 
conjugate  axis,  and  ef  the  transverse. 

6 


42 


AMERICAN  HOUSE-CARPENTER. 


113.  —  To  find  the  axis  and  base  of  the  parabola.  Let  m  l , 
{Fig-.  79,)  parallel  to  a  c,  be  the  direction  of  the  cutting  plane. 
From  m,  draw  m  d ,  at  right  angles  to  a  b  ;  then  l  m  will  be  the 
axis  and  height,  and  m  d  an  ordinate  and  half  the  base ;  as  at 
Fig.  92,  93. 

114.  —  To  find  the  height ,  base  and  transverse  axis  of  an 
hyperbola.  Let  o  r,  {Fig.  79,)  be  the  direction  of  the  cutting 
plane.  Extend  o  r  and  a  c  till  they  meet  at  n  ;  from  o,  draw 
o  p,  at  right  angles  to  a  b ;  then  r  o  will  be  the  height,  n  r  the 
transverse  axis,  and  o  p  half  the  base ;  as  at  Fig.  94. 

C 


115. — The  axis  being  given ,  to  find  the  foci,  and  to  describe 
an  ellipsis  with  a  strmg.  Let  a  b,  {Fig.  80,)  and  c  d,  be  the 
given  axes.  Upon  c,  with  a  e  or  be  for  radius,  describe  the  arc, 
ff;  then /and/,  the  points  at  which  the  arc  cuts  the  transverse 
axis,  will  be  the  foci.  At  /  and  /  place  two  pins,  and  another  at  c  ; 
tie  a  string  about  the  three  pins,  so  as  to  form  the  triangle,  ffc; 
remove  the  pin  from  c,  and  place  a  pencil  in  its  stead ;  keeping  the 
string  taut,  move  the  pencil  in  the  direction,  eg  a;  it  will  then 
describe  the  required  ellipsis.  The  lines,  fg  and  g  f  show  the 
position  of  the  string  when  the  pencil  arrives  at  g. 

This  method,  when  performed  correctly,  is  perfectly  accurate ; 
but  the  string  is  liable  to  stretch,  and  is,  therefore,  not  so  good  to 
use  as  the  trammel.  In  making  an  ellipse  by  a  string  or  twine, 
that  kind  should  be  used  which  has  the  least  tendency  to  elasticity. 
For  this  reason,  a  cotton  cord,  such  as  chalk-lines  are  commonly 
made  of,  is  not  proper  for  the  purpose  :  a  linen,  or  flaxen  cord  is 
much  better. 


PRACTICAL  GEOMETRY. 


43 


116. —  The  axes  being  given,  to  describe  an  ellipsis  with  a 
trammel.  Let  a  b  and  c  d ,  (Fig.  81,)  be  the  given  axes.  Place 
the  trammel  so  that  a  line  passing  through  the  centre  of  the 
grooves,  would  coincide  with  the  axes  ;  make  the  distance  from 
the  pencil,  e,  to  the  nut,/,  equal  to  half  c  d  ;  also,  from  the  pen¬ 
cil,  e,  to  the  nut,  g ,  equal  to  half  a  b  ;  letting  the  pins  under  the 
nuts  slide  in  the  grooves,  move  the  trammel,  e  g,  in  the  direction, 
c  b  d  ;  then  the  pencil  at  e  will  describe  the  required  ellipse. 

A  trammel  may  be  constructed  thus  :  take  two  straight  strips  of 
board,  and  make  a  groove  on  their  face,  in  the  centre  of  their 
width  ;  join  them  together,  in  the  middle  of  their  length,  at  right 
angles  to  one  another ;  as  is  seen  at  Fig.  81.  A  rod  is  then  to  be 
prepared,  having  two  moveable  nuts  made  of  wood,  with  a  mor¬ 
tice  through  them  of  the  size  of  the  rod,  and  pins  under  them 
large  enough  to  fill  the  grooves.  Make  a  hole  at  one  end  of  the 
rod,  in  which  to  place  a  pencil.  In  the  absence  of  a  regular  trarm 
mel,  a  temporary  one  may  be  made,  which,  for  any  short  job, 
will  answer  every  purpose.  Fasten  two  straight-edges  at  right 
angles  to  one  another.  Lay  them  so  as  to  coincide  with  the  axes 
of  the  proposed  ellipse,  having  the  angular  point  at  the  centre. 
Then,  in  a  rod  having  a  hole  for  the  pencil  at  one  end,  place  two 
brad-awls  at  the  distances  described  at  Art.  116.  While  the 
pencil  is  moved  in  the  direction  of  the  curve,  keep  the  brad-awls 
hard  against  the  straight-edges,  as  directed  for  using  the  tram¬ 
mel-rod,  and  one-quarter  of  the  ellipse  will  be  drawn.  Then, 
by  shifting  the  straight-edges,  the  other  three  quarters  in  succes¬ 
sion  may  be  drawn.  If  the  required  ellipse  be  not  too  large,  a 
carpenters’-square  may  be  made  use  of,  in  place  of  the  straight¬ 
edges. 

An  improved  method  of  constructing  the  trammel,  is  as  fol¬ 
lows  :  make  the  sides  of  the  grooves  bevilling  from  the  face  of 
the  stuff,  or  dove-tailing  instead  of  square.  Prepare  two  slips  of 
wood,  each  about  two  inches  long,  which  shall  be  of  a  shape  to 
just  fill  the  groove  when  slipped  in  at  the  end.  These,  instead  of 


44 


AMERICAN  HOUSE-CARPENTER. 


pins,  are  to  be  attached  one  to  each  of  the  moveable  nuts  with 
a  screw,  loose  enough  for  the  nut  to  move  freely  about  the  screw 
as  an  axis.  The  advantage  of  this  contrivance  is,  in  preventing 
the  nuts  from  slipping  out  of  their  places,  during  the  operation 
of  describing  the  curve. 


a- 


m 


i  d  i  m 


Fig.  82. 


117. —  To  describe  an  ellipsis  by  ordinates.  Let  a  b  and  c  d, 
(Fig.  82,)  be  given  axes.  With  a  e  or  e  b  for  radius,  de¬ 
scribe  the  quadrant, /g-  h ;  divide  f  h,  a  e  and  e  b,  each  into  a 
like  number  of  equal  parts,  as  at  1,  2  and  3  ;  through  these 
points,  draw  ordinates,  parallel  to  c  d  and  f  g  ;  take  the  distance, 
1  i,  and  place  it  at  1 1 ,  transfer  2  j  to  2  m ,  and  3  k  to  3  n;  through 
the  points,  a,  n,  m ,  l  and  c,  trace  a  curve,  and  the  ellipsis  will 
be  completed. 

The  greater  the  number  of  divisions  on  a  e,  &c.,  in  this  and 
the  following  problem,  the  more  points  in  the  curve  can  be  found, 
and  the  more  accurate  the  curve  can  be  traced.  If  pins  are 
placed  in  the  points,  n,  m ,  l ,  &c.,  and  a  thin  slip  of  wood  bent 
around  by  them,  the  curve  can  be  made  quite  correct.  This 
method  is  mostly  used  in  tracing  face-moulds  for  stair  hand- 
railing. 


c 


d 

Fig  83. 


118. —  To  describe  an  ellipsis  by  intersection  of  lines.  Let 


PRACTICAL  GEOMETRY.  45 

a  b  and  c  d,  {Fig.  83,)  be  given  axes.  Through  c,  draw  f  g, 
parallel  to  a  b  ;  from  a  and  b,  draw  a  f  and  b  g,  at  right  angles 
to  a  b  ;  divide  /  a,  g  b,  a  e  and  e  b,  each  into  a  like  number  of 
equal  parts,  as  at  1,  2,  3  and  o,  o,  o  ;  from  1,  2  and  3,  draw  lines 
to  c ;  through  o,  o  and  o,  draw  lines  from  d,  intersecting  those 
drawn  to  c  ;  then  a  curve,  traced  through  the  points,  i ,  i ,  i,  will 
be  that  of  an  ellipsis. 


Where  neither  trammel  nor  string  is  at  hand,  this,  perhaps,  is 
the  most  ready  method  of  drawing  an  ellipsis.  The  divisions 
should  be  small,  where  accuracy  is  desirable.  By  this  method, 
an  ellipsis  may  be  traced  without  the  axes,  provided  that  a  diame¬ 
ter  and  its  conjugate  be  given.  Thus,  a  b  and  c  d ,  {Fig.  84,)  are 
conjugate  diameters :  f  g  is  drawn  parallel  to  a  b,  instead  of 
being  at  right  angles  to  c  d  ;  also,  /  a  and  g  b  are  drawn  parallel 
to  c  d,  instead  of  being  at  right  angles  to  a  b. 


h  g  c  g  h 


119. —  To  describe  an  ellipsis  by  intersecting  arcs :  Let  a  b 


46 


AMERICAN  HOUSE-CARPENTER. 


and  c  d:  (Fig.  85,)  be  given  axes.  Between  one  of  the  foci,/ 
and  /  and  the  centre,  e,  mark  any  number  of  points,  at  random, 
as  1,  2  and  3  ;  upon  /and /,  with  b  1  for  radius,  describe  arcs  at 
g,  g,  g  and  g  ;  upon /  and /,  with  a  1  for  radius,  describe  arcs  inter¬ 
secting  the  others  at  g ,  g ,  g  and  g  ;  then  these  points  of  intersection 
will  be  in  the  curve  of  the  ellipsis.  The  other  points,  h  and  i,  are 
found  in  like  manner,  viz :  h  is  found  by  taking  b  2  for  one  radius, 
and  a  2  for  the  other ;  i  is  found  by  taking  b  3  for  one  radius,  and 
a  3  for  the  other,  always  using  the  foci  for  centres.  Then  by 
tracing  a  curve  through  the  points,  e,  g,  h,  i,  b,  &c.,  the  ellipse 
will  be  completed. 

This  problem  is  founded  upon  the  same  principle  as  that  of  the 
string.  This  is  obvious,  when  we  reflect  that  the  length  of  the 
string  is  equal  to  the  transverse  axis,  added  to  the  distance  between 
the  foci.  See  Fig.  80 ;  in  which  c  /  equals  a  e,  the  half  of  the 
transverse  axis. 


e 

Fig.  86. 


120. —  To  describe  a  figure  nearly  in  the  shape  of  an  ellip¬ 
sis ,  by  a  pair  of  compasses.  Let  a  b  and  c  d ,  (Fig.  86,)  be 
given  axes.  From  c,  draw  c  e,  parallel  to  a  b  ;  from  a ,  draw  a  e, 
parallel  t o  c  d;  join  e  and  d ;  bisect  e  a  in/;  join / and  c,  inter¬ 
secting  e  d  in  i;  bisect  i  c  in  o  ;  from  o,  draw  og,  at  right  angles 
to  i  c,  meeting  c  d  extended  to  g  ;  join  i  and  g ,  cutting  the  trans¬ 
verse  axis  in  r  ;  make  h  j  equal  to  h  g,  and  h  k  equal  to  hr; 
from/  through  r  and  k,  draw  j  m  and  7  n  ;  also,  from  g,  through 
k,  draw  g  l ;  upon  g  and  j,  with  g  c  for  radius,  describe  the 


PRACTICAL  GEOMETRY. 


47 


arcs,  i  l  and  m  n  ;  upon  r  and  k,  with  r  a  for  radius,  describe 
the  arcs,  m  i  and  l  n  ;  this  will  complete  the  figure. 

When  the  axes  are  proportioned  to  one  another  as  2  to  3,  the 
extremities,  c  and  d ,  of  the  shortest  axis,  will  be  the  centres  for 
describing  the  arcs,  i  l  and  m  n  ;  and  the  intersection  of  e  d  with 
the  transverse  axis,  will  be  the  centre  for  describing  the  arc,  m  i , 
&c.  As  the  elliptic  curve  is  continually  changing  its  course  from 
that  of  a  circle,  a  true  ellipsis  cannot  be  described  with  a  pair  of 
compasses.  The  above,  therefore,  is  only  an  approximation. 


c 


121. —  To  draw  an  oval  in  the  proportion ,  seven  by  nine. 
Let  c  d ,  {Fig.  87,)  be  the  given  conjugate  axis.  Bisect  c  d  in  o, 
and  through  o,  draw  a  6,  at  right  angles  to  c  d  ;  bisect  c  o  in  e  ; 
upon  o,  with  o  e  for  radius,  describe  the  circle,  e  f  g  h  ;  from  e, 
through  h  and  /  draw  e  j  and  e  i  ;  also,  from  g ,  through  h  and/, 
draw  g  k  and  g  l  ;  upon  g ,  with  g  c  for  radius,  describe  the  arc, 
kl ;  upon  e,  with  e  d  for  radius,  describe  the  arc,  j  i  ;  upon  h  and 
/  with  h  a  for  radius,  describe  the  arcs,  j  k  and  l  i;  this  will 
complete  the  figure. 

This  is  a  very  near  approximation  to  an  ellipsis ;  and  perhaps  no 
method  can  be  found,  by  which  a  well-shaped  oval  can  be  drawn 
with  greater  facility.  By  a  little  variation  in  the  process,  ovals 
of  different  proportions  may  be  obtained.  If  quarter  of  the  trans¬ 
verse  axis  is  taken  for  the  radius  of  the  circle,  efg  h ,  one  will  be 
drawn  in  the  proportion,  five  by  seven. 


48 


AMERICAN  HOUSE-CARPENTER. 


b 

Fig.  88. 


122. —  To  draw  a  tangent  to  an  ellipsis.  Let  abed ,  {Fig. 
88,)  be  the  given  ellipsis,  and  d  the  point  of  contact.  Find  the 
foci,  {Art.  115,)/ and/,  and  from  them,  through  d,  draw  fe  and 
f  d ;  bisect  the  angle,  {Art.  77,)  e  d  o,  with  the  line,  5  r  ;  then 
s  r  will  be  the  tangent  required. 


d 


c  Fig.  89. 


123.  — An  ellipsis  with  a  tangent  given,  to  detect  the  point 
of  contact.  Let  a  g  b f  {Fig.  89,)  be  the  given  ellipsis  and  tan¬ 
gent.  Through  the  centre,  e,  draw  a  b,  parallel  to  the  tangent ; 
any  where  between  e  and /  draw  c  d,  parallel  to  a  b  ;  bisect  cd  in 
o  ;  through  o  and  e,  draw  f  g  ;  then  g  will  be  the  point  of  con¬ 
tact  required. 

124.  — A  diameter  of  an  ellipsis  given ,  to  find  its  conjugate. 
Let  a  b,  {Fig.  89,)  be  the  given  diameter.  Find  the  line,/ g,  by 
the  last  problem ;  then  /  g  will  be  the  diameter  required. 


PRACTICAL  GEOMETRY. 


49 


125.  — Any  diameter  and  its  conjugate  being  given ,  to  as¬ 
certain  the  two  axes ,  and  thence  to  describe  the  ellipsis.  Let 
a  b  and  c  d ,  {Fig.  90,)  be  the  given  diameters,  conjugate  to  one 
another.  Through  c,  draw  e  f  parallel  to  a  b  ;  from  c,  draw  c 
g,  at  right  angles  to  e  f ;  make  c  g  equal  to  a  h  or  h  b  ;  join  g 
and  h  ;  upon  g,  with  g  c  for  radius,  describe  the  arc,  i  k  c  j  ; 
upon  h ,  with  the  same  radius,  describe  the  arc,  In;  through  the 
intersections,  l  and  n,  draw  n  o,  cutting  the  tangent,  ef  in  o  ; 
upon  o,  with  o  g  for  radius,  describe  the  semi-circle,  eigf ;  join 
e  andg-,  also  g  and/,  cutting  the  arc,  i  c  j ,  in  k  and  t ;  from  e, 
through  h,  draw  e  m,  also  from /  through  h,  draw  fp  ;  from  k 
and  t ,  draw  k  r  and  t  s,  parallel  to  g  /t,  cutting  e  m  in  r,  and  fp 
in  s  ;  make  h  m  equal  to  h  r,  and  h  p  equal  to  h  s  ;  then  r  m 
and  5  p  will  be  the  axes  required,  by  which  the  ellipsis  may  be 
drawn  in  the  usual  way. 

126.  —  To  describe  an  ellipsis ,  whose  axes  shall  be  propor¬ 
tionate  to  the  axes  of  a  larger  or  smaller  given  one.  Let  a 
cbd,  {Fig.  91,)  be  the  given  ellipsis  and  axes,  and  i  j  the  trans¬ 
verse  axis  of  a  proposed  smaller  one.  Join  a  and  c ;  from  i, 
draw  i  e,  parallel  to  a  c  ;  make  o  f  equal  to  o  e  ;  then  e  f  will  be 

7 


I 


AMERICAN  HOUSE-CARPENTER. 


50  . 


the  conjugate  axis  required,  and  will  bear  the  same  proportion  to 
i  j,  as  c  d  does  to  a  b.  (See  Art.  108.) 

12  3  l  3  2  1 


Fig.  92. 

127. — To  describe  a  parabola  by  intersection  of  lines.  Let 
m  l ,  (Fig.  92,)  be  the  axis  and  height,  (see  Fig.  79,)  and  d  d,  a 
double  ordinate  and  base  of  the  proposed  parabola.  Through  Z, 
draw  a  a,  parallel  to  d  d ;  through  d  and  eZ,  draw  d  a  and  d  a, 
parallel  to  ml ;  divide  a  d  and  d  m,  each  into  a  like  number  of 
equal  parts  ;  from  each  point  of  division  in  d  m ,  draw  the  lines, 
1  1,  2  2,  <kc.,  parallel  to  m  l ;  from  each  point  of  division  in  d 
a,  draw  lines  to  l ;  then  a  curve  traced  through  the  points  of 
intersection,  o,  o  and  o,  will  be  that  of  a  parabola. 

127,  a. — Another  method.  Let  m  Z,  [Fig.  93,)  be  the  axis  and 
height,  and  d  d  the  base.  Extend  m  Z,  and  make  Z  a  equal  to  m 
l  ;  join  a  and  d ,  and  a  and  d  ;  divide  a  d  and  a  d.  each  into  a 
like  number  of  equal  parts,  as  at  1,  2,  3,  &c. ;  join  1  and  1,  2  and 
2,  &c.,  and  the  parabola  will  be  completed. 


PRACTICAL  GEOMETRY. 


51 


128. — To  describe  an  hyperbola  by  intersection  of  lines. 

* 

Let  r  o,  {Fig.  94,)  be  the  height,  p  p  the  base,  and  n  r  the  trans¬ 
verse  axis.  (See  Fig.  79.)  Through  r,  draw  a  a,  parallel  to  p 
p  ;  from  p,  draw  a  p,  parallel  to  r  o ;  divide  a  p  and  p  o,  each 
into  a  like  number  of  equal  parts ;  from  each  of  the  points  of  di¬ 
visions  in  the  base,  draw  lines  to  n  ;  from  each  of  the  points  of 
division  in  a  p,  draw  lines  to  r  ;  then  a  curve  traced  through  the 
points  of  intersection,  o,  o,  &c.,  will  be  that  of  an  hyperbola. 

The  parabola  and  hyperbola  afford  handsome  curves  for  various 
mouldings. 


DEMONSTRATIONS. 


129. — To  impress  more  deeply  upon  the  mind  of  the  learner 
some  of  the  more  important  of  the  preceding  problems,  and  to 
indulge  a  very  common  and  praiseworthy  curiosity  to  discover 
the  cause  of  things,  are  some  of  the  reasons  why  the  following 
exercises  are  introduced.  In  all  reasoning,  definitions  are  ne¬ 
cessary  ;  in  order  to  insure,  in  the  minds  of  the  proponent  and 
respondent,  identity  of  ideas.  A  corollary  is  an  inference  deduced 
from  a  previous  course  of  reasoning.  An  axiom  is  a  proposition 
evident  at  first  sight.  In  the  following  demonstrations,  there  are 
many  axioms  taken  for  granted ;  (such  as,  things  equal  to  the 
same  thing  are  equal  to  one  another,  &c. ;)  these  it  was  thought 
not  necessary  to  introduce  in  form. 


Fig.  95. 


130. — Definition.  If  a  straight  line,  as  a  b,  {Fig.  95,)  stand 
upon  another  straight  line,  as  c  d ,  so  that  the  two  angles  made  at 


PRACTICAL  GEOMETRY. 


53 


the  point,  b,  are  equal — a  b  c  to  a  b  d,  (see  note  to  Art.  27,)  then 
each  of  the  two  angles  is  called  a  right  angle. 

131. — Definition.  The  circumference  of  every  circle  is  sup¬ 
posed  to  be  divided  into  360  equal  parts,  called  degrees  ;  hence 
a  semi-circle  contains  180  degrees,  a  quadrant  90,  &c. 


Fig.  96. 

132.  — Definition.  The  measure  of  an  angle  is  the  number  of 
degrees  contained  between  its  two  sides,  using  the  angular  point 
as  a  centre  upon  which  to  describe  the  arc.  Thus  the  arc,  c  e> 
(Fig.  96,)  is  the  measure  of  the  angle,  c  b  e  ;  e  a,  of  the  angle^ 
e  b  a  ;  and  a  d ,  of  the  angle,  ab  d. 

133.  — Corollary.  As  the  two  angles  at  b,  (Fig.  95,)  are  right 
angles,  and  as  the  semi-circle,  cad ,  contains  180  degrees,  ( Art. 
131,)  the  measure  of  two  right  angles,  therefore,  is  180  degrees  ; 
of  one  right  angle,  90  degrees  ;  of  half  a  right  angle,  45 ;  of 
one-third  of  a  right  angle,  30,  &.c. 

134.  — Definition.  In  measuring  an  angle,  (Art.  132,)  no  re¬ 
gard  is  to  be  had  to  the  length  of  its  sides,  but  only  to  the  degree 
of  their  inclination.  Hence  equal  angles  are  such  as  have  the 
same  degree  of  inclination,  without  regard  to  the  length  of  their 
sides. 


a  c 


135. — Axiom.  If  two  straight  lines,  parallel  to  one  another, 


54 


AMERICAN  HOUSE-CARPENTER. 


as  a  b  and  c  d ,  {Fig.  97,)  stand  upon  another  straight  line,  as  e /, 
the  angles,  a  b  f  and  c  d  /,  are  equal;  and  the  angle,  a  b  e,  is 
equal  to  the  angle,  c  d  e. 

136.  — Definition.  If  a  straight  line,  as  a  b,  {Fig.  96,)  stand 
obliquely  upon  another  straight  line,  as  c  d,  then  one  of  the  an¬ 
gles,  as  a  b  c,  is  called  an  obtuse  angle,  and  the  other,  as  ab  dy 
an  acute  angle. 

137.  — Axiom.  The  two  angles,  ab  d  and  ab  c,  {Fig.  96,)  are 
together  equal  to  two  right  angles,  {Art.  130,  133;)  also,  the 
three  angles,  a  b  d,e  b  a  and  cb  e,  are  together  equal  to  two  right 
angles. 

138.  — Corollary.  Hence  all  the  angles  that  can  be  made  upon 
one  side  of  a  line,  meeting  in  a  point  in  that  line,  are  together 
equal  to  two  right  angles. 

139.  — Corollary.  Hence  all  the  angles  that  can  be  made  on 
both  sides  of  a  line,  at  a  point  in  that  line,  or  all  the  angles  that 
can  be  made  about  a  point,  are  together  equal  to  four  right  angles. 


140. — Proposition.  If  to  each  of  two  equal  angles  a  third 
angle  be  added,  their  sums  will  be  equal.  Let  ab  c  and  d  ef, 
{Fig.  98,)  be  equal  angles,  and  the  angle,  i  j  k ,  the  one  to  be 
added.  Make  the  angles,  gb  a  and  hed ,  each  equal  to  the  given 
angle,  ijk;  then  the  angle,  gb  c,  will  be  equal  to  the  angle,  h  e 
f ;  for,  if  ab  c  and  d  e  f  be  angles  of  90  degrees,  and  i  j  k,  30, 
then  the  angles,  g  b  c  and  h  e  f,  will  be  each  equal  to  90  and 
30  added,  viz :  120  degrees. 


PRACTICAL  GEOMETRY. 


55 


a  d 


Fig.  99. 


141. — Proposition.  Triangles  that  have  two  of  their  sides 
and  the  angle  contained  between  them  respectively  equal,  have 
also  their  third  sides  and  the  two  remaining  angles  equal ;  and 
consequently  one  triangle  will  every  way  equal  the  other.  Let  a 
b  c ,  {Fig.  99,)  and  d  ef  be  two  given  triangles,  having  the  angle 
at  a  equal  to  the  angle  at  d,  the  side,  a  b,  equal  to  the  side,  d  e, 
and  the  side,  a  c,  equal  to  the  side,  df;  then  the  third  side  of 
one,  b  c,  is  equal  to  the  third  side  of  the  other,  e  f;  the  angle  at  b 
is  equal  to  the  angle  at  e ,  and  the  angle  at  c  is  equal  to  the  angle 
at/.  For,  if  one  triangle  be  applied  to  the  other,  the  three  points, 
b,  a,  c,  coinciding  with  the  three  points,  e,  d,  /  the  line,  b  c,  must 
coincide  with  the  line,  e  f;  the  angle  at  b  with  the  angle  at  e  ; 
the  angle  at  c  with  the  angle  at/;  and  the  triangle,  b  a  c,  be  every 
way  equal  to  the  triangle,  e  df. 


a 


Fig.  100. 

142. — Proposition.  The  two  angles  at  the  base  of  an  isoceles 
triangle  are  equal.  Let  a  b  c,  {Fig.  100,)  be  an  isoceles  triangle, 
of  which  the  sides,  a  b  and  a  c,  are  equal.  Bisect  the  angle,  {Art. 


56 


AMERICAN  HOUSE-CARPENTER. 


77,)  b  a  e,  by  the  line,  a  d.  Then  the  line,  b  a,  being  equal  to 
the  line,  a  c  ;  the  line,  a  d,  of  the  triangle,  A,  being  equal  to  the 
line,  a  d ,  of  the  triangle,  B,  being  common  to  each  ;  the  angle,  b 
a  d ,  being  equal  to  the  angle,  d  a  c  ;  the  line,  b  d,  must,  accord¬ 
ing  to  Art.  141,  be  equal  to  the  line,  d  c  ;  and  the  angle  at  b  must 
be  equal  to  the  angle  at  c. 


a  b  b  a 


143. — Proposition.  A  diagonal  crossing  a  parallelogram  di¬ 
vides  it  into  two  equal  triangles.  Let  abed ,  {Fig.  101,)  be  a 
given  parallelogram,  and  be,  a  line  crossing  it  diagonally.  Then, 
as  a  c  is  equal  to  b  d,  and  a  b  to  cd,  the  angle  at  a  to  the  angle 
at  d,  the  triangle,  A,  must,  according  to  Art.  141,  be  equal  to  the 
triangle,  B. 


h  b  a  Kb 


Fig.  102. 


144. — Proposition.  Let  abed,  {Fig.  102,)  be  a  given  pa¬ 
rallelogram,  and  be  a  diagonal.  At  any  distance  between  a  b  and 
c  d,  draw  e  /,  parallel  to  a  b  ;  through  the  point,  g,  the  intersection 
of  the  lines,  b  c  and  e  f,  draw  h  i,  parallel  to  b  d.  In  every  paral¬ 
lelogram  thus  divided,  the  parallelogram,  A,  is  equal  to  the  paral¬ 
lelogram,  B.  According  to  Art.  143,  the  triangle,  a  b  c,  is 
equal  to  the  triangle,  bed;  the  triangle,  C,  to  the  triangle,  D  ; 
and  E  to  F  ;  this  being  the  case,  take  D  and  F  from  the  triangle, 
bed,  and  C  and  E  from  the  triangle,  a  b  c,  and  what  remains 


PRACTICAL  GEOMETRY. 


67 


in  one  must  be  equal  to  what  remains  in  the  other ;  therefore,  the 
parallelogram,  A,  is  equal  to  the  parallelogram,  B. 


145.  — Proposition.  Parallelograms  standing  upon  the  same 
base  and  between  the  same  parallels,  are  equal.  Let  abed  and 
ef  cd,  (Fig.  103,)  be  given  parallelograms,  standing  upon  the 
same  base,  c  d:  and  between  the  same  parallels,  a  f  and  c  d. 
Then,  a  b  and  e  f  being  equal  to  c  d ,  are  equal  to  one  another ; 
b  e  being  added  to  both  a  b  and  e  /,  a  e  equals  b  f ;  the  line,  a  c , 
being  equal  to  b  d ,  and  a  e  to  b  f,  and  the  angle,  c  a  e,  being 
equal,  (Art.  135,)  to  the  angle,  d  6/,  the  triangle,  a  e  c,  must  be 
equal,  (Art.  141,)  to  the  triangle,  b  f  d  ;  these  two  triangles  being 
equal,  take  the  same  amount,  the  triangle,  beg,  from  each,  and 
what  remains  in  one,  a  b  g  c,  must  be  equal  to  what  remains  in 
the  other,  efdg  ;  these  two  quadrangles  being  equal,  add  the 
same  amount,  the  triangle,  c  g  d,  to  each,  and  they  must  still  be 
equal ;  therefore,  the  parallelogram,  abed,  is  equal  to  the  paral¬ 
lelogram,  efed. 

146.  — Corollary.  Hence,  if  a  parallelogram  and  triangle  stand 
upon  the  same  base  and  between  the  same  parallels,  the  parallelo¬ 
gram  will  be  equal  to  double  the  triangle.  Thus,  the  paral¬ 
lelogram,  a  d,  (Fig.  103,)  is  double,  (Art.  143,)  the  triangle, 
c  e  d. 

147.  — Proposition.  Let  abed,  (Fig.  104,)  be  a  given  quad¬ 
rangle  with  the  diagonal,  a  d.  From  6,  draw  b  e,  parallel  to  a  d; 
extend  c  d  to  e  ;  join  a  and  e  ;  then  the  triangle,  a  e  c,  will  be  equal 
in  area  to  the  quadrangle,  abed.  Since  the  triangles,  ad  b  and 
a  d  e,  stand  upon  the  same  base,  a  d,  and  between  the  same  paral- 

8 


58 


AMERICAN  HOUSE-CARPENTER. 


a 


lels,  a  d  and  b  e,  they  are  therefore  equal,  ( Art.  145,  146  ;)  and 
since  the  triangle,  C ,  is  common  to  both,  the  remaining  triangles,  A 
and  B ,  are  therefore  equal ;  then  B  being  equal  to  A,  the  triangle, 
a  e  c,  is  equal  to  the  quadrangle,  abed. 


148.  — Proposition.  If  two  straight  lines  cut  each  other,  as 
a  b  and  c  d,  (Fig.  105,)  the  vertical,  or  opposite  angles,  A  and 
C,  are  equal.  Thus,  a  e,  standing  upon  c  d,  forms  the  angles, 
B  and  C,  which  together  amount,  ( Art.  137,)  to  two  right  angles ; 
in  the  same  manner,  the  angles,  A  and  B ,  form  two  right  angles ; 
since  the  angles,  A  and  B,  are  equal  to  B  and  C,  take  the  same 
amount,  the  angle,  B ,  from  each  pair,  and  what  remains  of  one 
pair  is  equal  to  what  remains  of  the  other ;  therefore,  the  an¬ 
gle,  A,  is  equal  to  the  angle,  C.  The  same  can  be  proved  of 
the  opposite  angles,  B  and  D. 

149.  — Proposition.  The  three  angles  of  any  triangle  are 
equal  to  two  right  angles.  Let  a  b  c,  (Fig.  106,)  be  a  given  tri¬ 
angle,  with  its  sides  extended  to /,  e,  and  d,  and  the  line,  eg, 


PRACTICAL  GEOMETRY. 


59 


/  e 


drawn  parallel  to  be.  As  g  c  is  parallel  to  e  b,  the  angle,  g  c  d, 
is,  equal,  (Art.  135,)  to  the  angle,  ebd ;  as  the  lines,  f  c  and  b  e, 
cut  one  another  at  a,  the  opposite  angles,  f  a  e  and  b  a  c,  are 
equal,  (Art.  148  ;)  as  the  angle,/  a  e,  is  equal,  (Art.  135,)  to  the 
angle,  a  eg,  the  angle,  a  c  g,  is  equal  to  the  angle,  b  a  c  ;  there¬ 
fore,  the  three  angles  meeting  at  c,  are  equal  to  the  three  angles 
of  the  triangle,  a  b  c  ;  and  since  the  three  angles  at  c  are  equal, 
(Art.  137,)  to  two  right  angles,  the  three  angles  of  the  triangle,  a 
b  c,  must  likewise  be  equal  to  two  right  angles.  Any  triangle 
can  be  subjected  to  the  same  proof. 

150.  — Corollary.  Hence,  if  one  angle  of  a  triangle  be  a  right 
angle,  the  other  two  angles  amount  to  just  one  right  angle. 

151.  — Corollary.  If  one  angle  of  a  triangle  be  a  right  angle, 
and  the  two  remaining  angles  are  equal  to  one  another,  these  are 
each  equal  to  half  a  right  angle. 

152.  — Corollary.  If  any  two  angles  of  a  triangle  amount  to 
a  right  angle,  the  remaining  angle  is  a  right  angle. 

153.  — Corollary.  If  any  two  angles  of  a  triangle  are  together 
equal  to  the  remaining  angle,  that  remaining  angle  is  a  right 
angle. 

154.  — Corollary.  If  any  two  angles  of  a  triangle  are  each 
equal  to  two-thirds  of  a  right  angle,  the  remaining  angle  is  also 
equal  to  two-thirds  of  a  right  angle. 

155.  — Corollary.  Hence,  the  angles  of  an  equi-lateral  trian¬ 
gle,  are  each  equal  to  two-thirds  of  a  right  angle. 


60 


AMERICAN  HOUSE-CARPENTER. 


156. — Proposition.  If  from  the  extremities  of  the  diameter  of 
a  semi-circle,  two  straight  lines  be  drawn  to  any  point  in  the  cir¬ 
cumference,  the  angle  formed  by  them  at  that  point  will  be  a 
right  angle.  Let  a  b  c,  {Fig.  107,)  be  a  given  semi-circle  ;  and 
a  b  and  b  c,  lines  drawn  from  the  extremities  of  the  diameter,  a 
c,  to  the  given  point,  b  ;  the  angle  formed  at  that  point  by  these 
lines,  is  a  right  angle.  Join  the  point,  b ,  and  the  centre,  d  ;  the 
lines,  d  a,  d  b  and  d  c,  being  radii  of  the  same  circle,  are  equal ; 
the  angle  at  a  is  therefore  equal,  {Art.  142,)  to  the  angle,  a  b  d, 
also,  the  angle  at  c  is,  for  the  same  reason,  equal  to  the  angle,  d  b 
c  ;  the  angle,  a  b  c,  being  equal  to  the  angles  at  a  and  c  taken 
together,  must  therefore,  {Art.  152,)  be  a  right  angle. 


157. — Proposition.  The  square  of  the  hypothenuse  of  a 
right-angled  triangle,  is  equal  to  the  squares  of  the  two  remaining 
sides.  Let  a  b  c,  {Fig.  108,)  be  a  given  right-angled  triangle, 
having  a  square  formed  on  each  of  its  sides :  then,  the  square,  b  e,  is 
equal  to  the  squares,  h  c  and  g  b,  taken  together.  This  can  be* 


i 


PRACTICAL  GEOMETRY. 


61 


proved  by  showing  that  the  parallelogram,  b  l ,  is  equal  to  the  square, 
gb  ;  and  that  the  parallelogram,  c  Z,  is  equal  to  the  square,  h  c.  The 
angle,  c  b  d,  is  a  right  angle,  and  the  angle,  abf,  is  aright  angle  ; 
add  to  each  of  these  the  angle,  ab  c  ;  then  the  angle,/  b  c,  will  evi¬ 
dently  be  equal,  {Art.  140,)  to  the  angle,  ab  d  ;  the  triangle,  /  b  c, 
and  the  square,  g  6,  being  both  upon  the  same  base,/Z>,  and  between 
the  same  parallels,  /  b  andg-  c,  the  square,  g  b:  is  equal,  {Art.  146,) 
to  twice  the  triangle,  f  b  c  ;  the  triangle,  ab  d,  and  the  parallelo¬ 
gram,  b  Z,  being  both  upon  the  same  base,  b  d,  and  between  the 
same  parallels,  b  d  and  a  Z,  the  parallelogram,  b  Z,  is  equal  to  twice 
the  triangle,  a  b  d ;  the  triangles,/  b  c  and  a  b  d,  being  equal  to 
one  another,  ( Art.  141,)  the  square,  g  b,  is  equal  to  the  parallelo¬ 
gram,  b  Z,  either  being  equal  to  twice  the  triangle,/ b  c  or  a  b  d. 
The  method  of  proving  h  c  equal  to  c  Z  is  exactly  similar — thus 
proving  the  square,  b  e,  equal  to  the  squares,  h  c  and  g  Z>,  taken 
together. 

This  problem,  which  is  the  47th  of  the  First  Book  of  Euclid, 
is  said  to  have  been  demonstrated  first  by  Pythagoras.  It  is  sta¬ 
ted,  (but  the  story  is  of  doubtful  authority,)  that  as  a  thank-offer¬ 
ing  for  its  discovery  he  sacrificed  a  hundred  oxen  to  the  gods. 
From  this  circumstance,  it  is  sometimes  called  the  hecatomb  pro¬ 
blem.  It  is  of  great  value  in  the  exact  sciences,  more  especially 
in  Mensuration  and  Astronomy,  in  which  many  otherwise  intri¬ 
cate  calculations  are  by  it  made  easy  of  solution. 

These  demonstrations,  which  relate  mostly  to  the  problems  pre¬ 
viously  given,  are  introduced  to  satisfy  the  learner  in  regard  to 
their  mathematical  accuracy.  By  studying  and  thoroughly  un¬ 
derstanding  them,  he  will  soonest  arrive  at  a  knowledge  of  their 
importance,  and  be  likely  the  longer  to  retain  them  in  memory. 
Should  he  have  a  relish  for  such  exercises,  and  wish  to  continue 
them  farther,  he  may  consult  Euclid’s  Elements,  in  which  the 
whole  subject  of  theoretical  geometry  is  treated  of  in  a  manner 
sufficiently  intelligible  to  be  understood  by  the  young  mechanic. 


62 


AMERICAN  HOUSE-CARPENTER. 


The  house- carpenter,  especially,  needs  information  of  this  kind, 
and  were  he  thoroughly  acquainted  with  the  principles  of  geome¬ 
try,  he  would  be  much  less  liable  to  commit  mistakes,  and  be 
better  qualified  to  excel  in  the  execution  of  his  often  difficult  un¬ 
dertakings. 


SECTION  II.— ARCHITECTURE. 


HISTORY  OF  ARCHITECTURE. 

158.  — Architecture  has  been  defined  to  be — “  the  art  of  build¬ 
ing  ;”  but,  in  its  common  acceptation,  it  is — “  the  art  of  designing 
and  constructing  buildings,  in  accordance  with  such  principles  as 
constitute  stability,  utility  and  beauty.”  The  literal  signification 
of  the  Greek  word  arclii-tecton ,  from  which  the  word  architect 
is  derived,  is  chief-carpenter ;  but  the  architect  has  always  been 
known  as  the  chief  designer  rather  than  the  chief  builder.  Of 
the  three  classes  into  which  architecture  has  been  divided — viz., 
Civil,  Military,  and  Naval,  the  first  is  that  which  refers  to  the 
construction  of  edifices  known  as  dwellings,  churches  and  other 
public  buildings,  bridges,  &c.,  for  the  accommodation  of  civilized 
man — and  is  the  subject  of  the  remarks  which  follow. 

159.  — This  is  one  of  the  most  ancient  of  the  arts :  the  scrip¬ 
tures  inform  us  of  its  existence  at  a  very  early  period.  Cain, 
the  son  of  Adam, — “  budded  a  city,  and  called  the  name  of  the 
city  after  the  name  of  his  son,  Enoch” — but  of  the  peculiar  style 
or  manner  of  building  we  are  not  informed.  It  is  presumed  that 
it  was  not  remarkable  for  beauty,  but  that  utility  and  perhaps  sta¬ 
bility  were  its  characteristics.  Soon  after  the  deluge — that  me- 


64 


AMERICAN  HOUSE-CARPENTER. 


morable  event,  which  removed  from  existence  all  traces  of  the 
works  of  man — the  Tower  of  Babel  was  commenced.  This  was 
a  work  of  such  magnitude  that  the  gathering  of  the  materials, 
according  to  some  writers,  occupied  three  years  ;  the  period  from 
its  commencement  until  the  work  was  abandoned,  was  twenty- 
two  years ;  and  the  bricks  were  like  blocks  of  stone,  being  twenty 
feet  long,  fifteen  broad  and  seven  thick.  Learned  men  have  given 
it  as  their  opinion,  that  the  tower  in  the  temple  of  Belus  at  Baby¬ 
lon  was  the  same  as  that  which  in  the  scriptures  is  called  the 
Tower  of  Babel.  The  tower  of  the  temple  of  Belus  was  square 
at  its  base,  each  side  measuring  one  furlong,  and  consequently 
half  a  mile  in  circumference.  Its  form  was  that  of  a  pyramid 
and  its  height  was  660  feet.  It  had  a  winding  passage  on  the 
outside  from  the  base  to  the  summit,  which  was  wide  enough  for 
two  carriages. 

160.  — Historical  accounts  of  ancient  cities,  of  which  there  are 
now  but  few  remains — such  as  Babylon,  Palmyra  and  Ninevah 
of  the  Assyrians ;  Sidon,  Tyre,  Aradus  and  Serepta  of  the  Phoe¬ 
nicians  ;  and  Jerusalem,  with  its  splendid  temple,  of  the  Israelites 
— show  that  architecture  among  them  had  made  great  advances. 
Ancient  monuments  of  the  art  are  found  also  among  other  nations  ? 
the  subterraneous  temples  of  the  Hindoos  upon  the  islands,  Ele- 
phanta  and  Salsetta  ;  the  ruins  of  Persepolis  in  Persia ;  pyramids, 
obelisks,  temples,  palaces  and  sepulchres  in  Egypt — all  prove  that 
the  architects  of  those  early  times  were  possessed  of  skill  and 
judgment  highly  cultivated.  The  principal  characteristics  of 
their  works,  are  gigantic  dimensions,  immoveable  solidity,  and,  in 
some  instances,  harmonious  splendour.  The  extraordinary  size 
of  some  is  illustrated  in  the  pyramids  of  Egypt.  The  largest  of 
these  stands  not  far  from  the  city  of  Cairo :  its  base,  which  is 
square,  covers  about  11|  acres,  and  its  height  is  nearly  500  feet. 
The  stones  of  which  it  is  built  are  immense — the  smallest  being 
full  thirty  feet  long. 

161.  — Among  the  Greeks,  architecture  was  cultivated  as  a  fine 


ARCHITECTURE. 


65 


art,  and  rapidly  advanced  towards  perfection.  Dignity  and  grace 
were  added  to  stability  and  magnificence.  In  the  Doric  order, 
their  first  style  of  building,  this  is  fully  exemplified.  Phidias, 
Ictinus  and  Callicrates,  are  spoken  of  as  masters  in  the  art  at  this 
period;  the  encouragement  and  support  of  Pericles  stimulated 
them  to  a  noble  emulation.  The  beautiful  temple  of  Minerva, 
erected  upon  the  acropolis  of  Athens,  the  Propyleum,  the  Odeum 
and  others,  were  lasting  monuments  of  their  success.  The  Ionic 
and  Corinthian  orders  were  added  to  the  Doric,  and  many  mag¬ 
nificent  edifices  arose.  These  exemplified,  in  their  chaste  propor¬ 
tions,  the  elegant  refinement  of  Grecian  taste.  Improvement  in 
Grecian  architecture  continued  to  advance,  until  perfection  seems 
to  have  been  attained.  The  specimens  which  have  been  partially 
preserved,  exhibit  a  combination  of  elegant  proportion,  dignified 
simplicity  and  majestic  grandeur.  Architecture  among  the 
Greeks  was  at  the  height  of  its  glory  at  the  period  immediately 
preceding  the  Peloponnesian  war ;  after  which  the  art  declined. 
An  excess  of  enrichment  succeeded  its  former  simple  grandeur ; 
yet  a  strict  regularity  was  maintained  amid  the  profusion  of  orna¬ 
ment.  After  the  death  of  Alexander,  323  B.  C.,  a  love  of  gaudy 
splendour  increased :  the  consequent  decline  of  the  art  was 
visible,  and  the  Greeks  afterwards  paid  but  little  attention  to  the 
science. 

162. — While  the  Greeks  were  masters  in  architecture,  which 
they  applied  mostly  to  their  temples  and  other  public  buildings, 
the  Romans  gave  their  attention  to  the  science  in  the  construction 
of  the  many  aqueducts  and  sewers  with  which  Rome  abounded ; 
building  no  such  splendid  edifices  as  adorned  Athens,  Corinth 
and  Ephesus,  until  about  200  years  B.  C.,  when  their  intercourse 
with  the  Greeks  became  more  extended.  Grecian  architecture 
was  introduced  into  Rome  by  Sylla ;  by  whom,  as  also  by  Marius 
and  Caesar,  many  large  edifices  were  erected  in  various  cities  of 
Italy.  But  under  Caesar  Augustus,  at  about  the  beginning  of  the 
Christian  era,  the  art  arose  to  the  greatest  perfection  it  ever  at- 

9 


66 


AMERICAN  HOUSE-CARPENTER. 


tained  in  Italy.  Under  his  patronage,  Grecian  artists  were  en¬ 
couraged,  and  many  emigrated  to  Rome.  It  was  at  about  this 
time  that  Solomon’s  temple  at  Jerusalem  was  rebuilt  by  Herod — 
a  Roman.  This  was  46  years  in  the  erection,  and  was  most  pro¬ 
bably  of  the  Grecian  style  of  building — perhaps  of  the  Corin¬ 
thian  order.  Some  of  the  stones  of  which  it  was  built  were  46 
feet  long,  21  feet  high  and  14  thick ;  and  others  were  of  the 
astonishing  length  of  82  feet.  The  porch  rose  to  a  great  height ; 
the  whole  being  built  of  white  marble  exquisitely  polished.  This 
is  the  building  concerning  which  it  was  remarked — “  Master,  see 
what  manner  of  stones,  and  what  buildings  are  here.”  For  the 
construction  of  private  habitations  also,  finished  artists  were  em¬ 
ployed  by  the  Romans  :  their  dwellings  being  often  built  with  the 
finest  marble,  and  their  villas  splendidly  adorned.  After  Augus¬ 
tus,  his  successors  continued  to  beautify  the  city,  until  the  reign  of 
Constantine ;  who,  having  removed  the  imperial  residence  to 
Constantinople,  neglected  to  add  to  the  splendour  of  Rome  ;  and 
the  art,  in  consequence,  soon  fell  from  its  high  excellence. 

Thus  we  find  that  Rome  was  indebted  to  Greece  for  what  she 
possessed  of  architecture — not  only  for  the  knowledge  of  its  prin¬ 
ciples,  but  also  for  many  of  the  best  buildings  themselves  ;  these 
having  been  originally  erected  in  Greece,  and  stolen  by  the  un¬ 
principled  conquerors — taken  down  and  removed  to  Rome. 
Greece  was  thus  robbed  of  her  best  monuments  of  architecture. 
Touched  by  the  Romans,  Grecian  architecture  lost  much  of  its 
elegance  and  dignity.  The  Romans,  though  justly  celebrated 
for  their  scientific  knowledge  as  displayed  in  the  construction  of 
their  various  edifices,  were  not  capable  of  appreciating  the  simple 
grandeur,  the  refined  elegance  of  the  Grecian  style ;  but  sought 
to  improve  upon  it  by  the  addition  of  luxurious  enrichment,  and 
thus  deprived  it  of  true  elegance.  In  the  days  of  Nero,  whose 
palace  of  gold  is  so  celebrated,  buildings  were  lavishly  adorned. 
Adrian  did  much  to  encourage  the  art ;  but  not  satisfied  with  the 
simplicity  of  the  Grecian  style,  the  artists  of  his  time  aimed  at 


ARCHITECTURE. 


67 


inventing  new  ones,  and  added  to  the  already  redundant  embel¬ 
lishments  of  the  previous  age.  Hence  the  origin  of  the  pedestal, 
the  great  variety  of  intricate  ornaments,  the  convex  frieze,  the 
round  and  the  open  pediments,  &c.  The  rage  for  luxury 
continued  until  Alexander  Severus,  who  made  some  improve¬ 
ment  ;  but  very  soon  after  his  reign,  the  art  began  rapidly  to 
decline,  as  particularly  evidenced  in  the  mean  and  trifling  charac¬ 
ter  of  the  ornaments. 

163. — The  Goths  and  Vandals,  when  they  overran  the  coun¬ 
tries  of  Italy,  Greece,  Asia  and  Africa,  destroyed  most  of  the 
works  of  ancient  architecture.  Cultivating  no  art  but  that  of 
war,  these  savage  hordes  could  not  be  expected  to  take  any  interest 
in  the  beautiful  forms  and  proportions  of  their  habitations.  From 
this  time,  architecture  assumed  an  entirely  different  aspect.  The 
celebrated  styles  of  Greece  were  unappreciated  and  forgotten;  and 
modern  architecture  took  its  first  step  on  the  platform  of  existence. 
The  Goths,  in  their  conquering  invasions,  gradually  extended  it 
over  Italy,  France,  Spain,  Portugal  and  Germany,  into  England. 
From  the  reign  of  Gallienus  may  be  reckoned  the  total  extinction 
of  the  arts  among  the  Romans.  From  his  time  until  the  6th  or 
7th  century,  architecture  was  almost  entirely  neglected.  The 
buildings  which  were  erected  during  this  suspension  of  the  arts, 
were  very  rude.  Being  constructed  of  the  fragments  of  the  edi¬ 
fices  which  had  been  demolished  by  the  Visigoths  in  their  unre¬ 
strained  fury,  and  the  builders  being  destitute  of  a  proper  know¬ 
ledge  of  architecture,  many  sad  blunders  and  extensive  patch- 
work  might  have  been  seen  in  their  construction — entablatures 
inverted,  columns  standing  on  their  wrong  ends,  and  other  ridi¬ 
culous  arrangements  characterized  their  clumsy  work.  The  vast 
number  of  columns  which  the  ruins  around  them  afforded,  they 
used  as  piers  in  the  construction  of  arcades — which  by  some  is 
thought,  after  having  passed  through  various  changes,  to  have 
been  the  origin  of  the  plan  of  the  Gothic  cathedral.  Buildings 
generally,  which  are  not  of  the  classical  styles,  and  which  were 


68 


AMERICAN  HOUSE-CARPENTER. 


erected  after  the  fall  of  the  Roman  empire,  have  by  some  been 
indiscriminately  included  under  the  term  Gothic .  But  the 
changes  which  architecture  underwent  during  the  dark  ages,  show 
that  there  were  several  distinct  modes  of  building. 

164.  — Theodoric,  king  of  the  Ostrogoths,  a  friend  of  the  arts, 
who  reigned  in  Italy  from  A.  D.  493  to  525,  endeavoured  to  re¬ 
store  and  preserve  some  of  the  ancient  buildings  ;  and  erected 
others,  the  ruins  of  which  are  still  seen  at  Verona  and  Ravenna. 
Simplicity  and  strength  are  the  characteristics  of  the  structures 
erected  by  him ;  they  are,  however,  devoid  of  grandeur  and  ele¬ 
gance,  or  fine  proportions.  These  are  properly  of  the  Gothic 
style ;  by  some  called  the  old  Gothic  to  distinguish  it  from  the 
pointed  style,  which  is  generally  called  modern  Gothic. 

165. — The  Lombards,  who  ruled  in  Italy  from  A.  D.  568,  had 
no  taste  for  architecture  nor  respect  for  antiquities.  Accordingly, 
they  pulled  down  the  splendid  monuments  of  classic  architecture 
which  they  found  standing,  and  erected  in  their  stead  huge  build¬ 
ings  of  stone  which  were  greatly  destitute  of  proportion,  elegance 
or  utility — their  characteristics  being  scarcely  any  thing  more  than 
stability  and  immensity  combined  with  ornaments  of  a  puerile  cha¬ 
racter.  Their  churches  were  disfigured  with  rows  of  small  columns 
along  the  cornice  of  the  pediment,  small  doors  and  windows  with 
circular  heads,  roofs  supported  by  arches  having  arched  buttresses 
to  resist  their  thrust,  and  a  lavish  display  of  incongruous  orna¬ 
ments.  This  kind  of  architecture  is  called,  the  Lombard  style, 
and  was  employed  in  the  7th  century  in  Pavia,  the  chief  city  of 
the  Lombards ;  at  which  city,  as  also  at  many  other  places,  a 
great  many  edifices  were  erected  in  accordance  with  its  inelegant 
forms.  • 

166.  — The  Byzantine  architects,  from  Byzantium,  Constantino¬ 
ple,  erected  many  spacious  edifices ;  among  which  are  included 
the  cathedrals  of  Bamberg,  Worms  and  Mentz,  and  the  most  an 
cient  part  of  the  minster  at  Strasburg  ;  in  all  of  these  they  com¬ 
bined  the  Roman-Ionic  order  with  the  Gothic  of  the  Lombards. 


ARCHITECTURE. 


69 


This  style  is  called  the  Lombard-Byzantine.  To  the  last  style 
there  were  afterwards  added  cupolas  similar  to  those  used  in  the 
east,  together  with  numerous  slender  pillars  with  tasteless  capi¬ 
tals,  and  the  many  minarets  which  are  the  characteristics  of  the 
proper  Byzantine ,  or  Oriental  style. 

167.  — In  the  eighth  century,  when  the  Arabs  and  Moors  de¬ 
stroyed  the  kingdom  of  the  Goths,  the  arts  and  sciences  were 
mostly  in  possession  of  the  Musselmen-conquerors ;  at  which 
time  there  were  three  kinds  of  architecture  practised ;  viz  :  the 
Arabian,  the  Moorish  and  the  modern-Gothic.  The  Arabian 
style  was  formed  from  Greek  models,  having  circular  arches 
added,  and  towers  which  terminated  with  globes  and  minarets. 
The  Moorish  is  very  similar  to  the  Arabian,  being  distinguished 
from  it  by  arches  in  the  form  of  a  horse-shoe.  It  originated  in 
Spain  in  the  erection  of  buildings  with  the  ruins  of  Roman  archi¬ 
tecture,  and  is  seen  in  all  its  splendour  in  the  ancient  palace  of  the 
Mohammedan  monarchs  at  Grenada,  called  the  Alhambra ,  or  red- 
house.  The  Modern-Gothic  was  originated  by  the  Visigoths 
in  Spain  by  a  combination  of  the  Arabian  and  Moorish  styles ; 
and  introduced  by  Charlemagne  into  Germany.  On  account  of 
the  changes  and  improvements  it  there  underwent,  it  was,  at  about 
the  13th  or  14th  century,  termed  the  German ,  or  romantic  style. 
It  is  exhibited  in  great  perfection  in  the  towers  of  the  minster  of 
Strasburgh,  the  cathedral  of  Cologne  and  other  edifices.  The 
most  remarkable  features  of  this  lofty  and  aspiring  style,  are  the 
lancet  or  pointed  arch,  clustered  pillars,  lofty  towers  and  flying 
buttresses.  It  was  principally  employed  in  ecclesiastical  archi¬ 
tecture,  and  in  this  capacity  introduced  into  France,  Italy,  Spain, 
and  England. 

168.  — The  Gothic  architecture  of  England  is  divided  into  the 
Norman ,  the  Early-English ,  the  Decorated ,  and  the  Perpen¬ 
dicular  styles.  The  Norman  is  principally  distinguished  by  the 
character  of  its  ornaments — the  chevron,  or  zigzag ,  being  the 
most  common.  Buildings  in  this  style  were  erected  in  the  12th 


70 


AMERICAN  HOUSE-CARPENTER. 


century.  The  Early-English  is  celebrated  for  the  beauty  of  its 
edifices,  the  chaste  simplicity  and  purity  of  design  which  they 
display,  and  the  peculiarly  graceful  character  of  its  foliage.  This 
style  is  of  the  13th  century.  The  Decorated  style,  as  its  name 
implies,  is  characterized  by  a  great  profusion  of  enrichment, 
which  consists  principally  of  the  crocket,  or  feathered-ornament, 
and  ball-flower.  It  was  mostly  in  use  in  the  14th  century.  The 
Perpendicular  style,  which  dates  from  the  15th  century,  is  distin¬ 
guished  by  its  high  towers,  and  parapets  surmounted  with  spires 
similar  in  number  and  grouping  to  oriental  minarets. 

169. — Thus  these  several  styles,  which  have  been  erroneously 
termed  Gothic ,  were  distinguished  by  peculiar  characteristics  as  well 
as  by  different  names.  The  first  symptoms  of  a  desire  to  return  to  a 
pure  style  in  architecture,  after  the  ruin  caused  by  the  Goths,  was 
manifested  in  the  character  of  the  art  as  displayed  in  the  church 
of  St.  Sophia  at  Constantinople,  which  was  erected  by  Justinian 
in  the  6th  century.  The  church  of  St.  Mark  at  Venice,  which 
arose  in  the  10th  or  11th  century,  was  the  work  of  Grecian  archi¬ 
tects,  and  resembles  in  magnificence  the  forms  of  ancient  archi¬ 
tecture.  The  cathedral  at  Pisa,  a  wonderful  structure  for  the  age, 
was  erected  by  a  Grecian  architect  in  1016.  The  marble  with 
Avhich  the  walls  of  this  building  Avere  faced,  and  of  Avhich  the  four 
rows  of  columns  that  support  the  roof  are  composed,  is  said  to  be 
of  an  excellent  character.  The  Campanile,  or  leaning-tOAver  as  it 
is  usually  called,  Avas  erected  near  the  cathedral  in  the  1.2th  cen¬ 
tury.  Its  inclination  is  generally  supposed  to  have  arisen  from 
a  poor  foundation ;  although  by  some  it  is  said  to  have  been  thus 
constructed  originally,  in  order  to  inspire  in  the  minds  of  the 
beholder  sensations  of  sublimity  and  awe.  In  the  13th  century, 
the  science  in  Italy  Avas  slowly  progressing  ;  many  fine  churches 
Avere  erected,  the  style  of  Avhich  displayed  a  decided  ad\7ance  in 
the  progress  tOAvards  pure  classical  architecture.  In  other  parts 
of  Europe,  the  Gothic,  or  pointed  style,  was  prevalent.  The 
cathedral  at  Strasburg,  designed  by  Irwin  Steinbeck,  was  erected 


ARCHITECTURE. 


71 


in  the  13th  and  14th  centuries.  In  France  and  England  during 
the  14th  century,  many  very  superior  edifices  were  erected  in  this 
style. 

170.  — In  the  14th  and  15th  centuries,  and  particularly  in  the 
latter,  architecture  in  Italy  was  greatly  revived.  The  masters  began 
to  study  the  remains  of  ancient  Roman  edifices  ;  and  many  splen¬ 
did  buildings  were  erected,  which  displayed  a  purer  taste  in  the 
science.  Among  others,  St.  Peter’s  of  Rome,  which  was  built 
about  this  time,  is  a  lasting  monument  of  the  architectural  skill  of 
the  age.  Giocondo,  Michael  Angelo,  Palladio,  Yignola,  and  other 
celebrated  architects,  each  in  their  turn,  did  much  to  restore  the  art 
to  its  former  excellence.  In  the  edifices  which  were  erected  under 
their  direction,  however,  it  is  plainly  to  be  seen  that  they  studied 
not  from  the  pure  models  of  Greece,  but  from  the  remains  of  the 
deteriorated  architecture  of  Rome.  The  high  pedestal,  the  cou¬ 
pled  columns,  the  rounded  pediment,  the  many  curved-and-t  wisted 
enrichments,  and  the  convex  frieze,  were  unknown  to  pure  Gre¬ 
cian  architecture.  Yet  their  efforts  were  serviceable  in  correcting, 
to  a  good  degree,  the  very  impure  taste  that  had  prevailed  since 
the  overthrow  of  the  Roman  empire. 

171.  — At  about  this  time,  the  Italian  masters  and  numerous 
artists  who  had  visited  Italy  for  the  purpose,  spread  the  Roman 
style  over  various  countries  of  Europe  ;  which  was  gradually  re¬ 
ceived  into  favor  in  place  of  the  modern-Gothic.  This  fell  into 
disuse ;  although  it  has  of  late  years  been  again  cultivated.  It 
requires  a  building  of  great  magnitude  and  complexity  for  a  per¬ 
fect  display  of  its  beauties.  In  America  at  the  present  time,  the 
pure  Grecian  style  is  more  or  less  studied  ;  and  perhaps  the  sim¬ 
plicity  of  its  principles  is  better  adapted  to  a  republican  country, 
than  the  intricacy  and  extent  of  those  of  the  Gothic. 

STYLES  OF  ARCHITECTURE. 

172.  — It  is  generally  acknowledged  that  the  various  styles  in 
architecture,  were  originated  in  accordance  with  the  different  pur- 


72 


AMERICAN  HOUSE-CARPENTER. 


suits  of  the  early  inhabitants  of  the  earth ;  and  were  brought  by 
their  descendants  to  their  present  state  of  perfection,  through  the 
propensity  for  imitation  and  desire  of  emulation  which  are  found 
more  or  less  among  all  nations.  Those  that  followed  agricultural 
pursuits,  from  being  employed  constantly  upon  the  same  piece  of 
land,  needed  a  permanent  residence,  and  the  wopden  hut  was  the 
offspring  of  their  wants  ;  while  the  shepherd,  who  followed  his 
flocks  and  was  compelled  to  traverse  large  tracts  of  country  for 
pasture,  found  the  tent  to  be  the  most  portable  habitation  ;  again, 
the  man  devoted  to  hunting  and  fishing — an  idle  and  vagabond 
way  of  living — is  naturally  supposed  to  have  been  content  with 
the  cavern  as  a  place  of  shelter.  The  latter  is  said  to  have  been 
the  origin  of  the  Egyptian  style ;  while  the  curved  roof  of  Chi¬ 
nese  structures  gives  a  strong  indication  of  their  having  had  the 
tent  for  their  model ;  and  the  simplicity  of  the  original  style  of 
the  Greeks,  (the  Doric,)  shows  quite  conclusively,  as  is  generally 
conceded,  that  its  original  was  of  wood.  The  modern-Gothifi,  or 
pointed  style,  which  was  most  generally  confined  to  ecclesiastical 
structures,  is  said  by  some  to  have  originated  in  an  attempt  to 
imitate  the  bower,  or  grove  of  trees,  in  which  the  ancients  per¬ 
formed  their  idol-worship. 

173.  — There  are  numerous  styles,  or  orders,  in  architecture  ; 
and  a  knowledge  of  the  peculiarities  of  each,  is  important  to  the 
student  in  the  art.  The  Stylobate  is  the  substructure,  or  base- 
ment,  upon  which  the  columns  of  an  order  are  arranged.  In 
Roman  architecture — especially  in  the  interior  of  an  edifice — it 
frequently  occurs  that  each  column  has  a  separate  substructure; 
this  is  called  a  pedestal.  If  possible,  the  pedestal  should  be 
avoided  in  all  cases  ;  because  it  gives  to  the  column  the  appear¬ 
ance  of  having  been  originally  designed  for  a  small  building, 
and  afterwards  pieced-out  to  make  it  long  enough  for  a  larger 
one. 

174.  — An  Order,  in  architecture,  is  composed  of  two  princi¬ 
pal  parts,  viz :  the  column  and  the  entablature. 


ARCHITECTURE. 


73 


175.  — The  Column  is  composed  of  the  base,  shaft  and  capital. 

176.  — The  Entablature,  above  and  supported  by  the 
columns,  is  horizontal ;  and  is  composed  of  the  architrave,  frieze 
and  cornice.  These  principal  parts  are  again  divided  into  various 
members  and  mouldings.  (See  Sect.  III.) 

177.  — The  Base  of  a  column  is  so  called  from  basis ,  a  founda¬ 
tion,  or  footing. 

178.  — The  Shaft,  the  upright  part  of  a  column  standing  upon 
the  base  and  crowned  with  the  capital,  is  from  shafto ,  to  dig — 
in  the  manner  of  a  well,  whose  inside  is  not  unlike  the  form  of  a 
column. 

179.  — The  Capital,  from  kephale  or  caput ,  the  head,  is  the 
uppermost  and  crowning  part  of  the  column. 

180.  — The  Architrave,  from  archly  chief  or  principal,  and 
trahs ,  a  beam,  is  that  part  of  the  entablature  which  lies  in  imme¬ 
diate  connection  with  the  column. 

181.  — The  Frieze,  from  fibron^  a  fringe  or  border,  is  that  part 
of  the  entablature  which  is  immediately  above  the  architrave  and 
beneath  the  cornice.  It  was  called  by  some  of  the  ancients, 
zophorus ,  because  it  was  usually  enriched  with  sculptured 
animals. 

182.  — The  Cornice,  from  corona ,  to  crown,  is  the  upper  and 
projecting  part  of  the  entablature — being  also  the  uppermost  and 
crowning  part  of  the  whole  order. 

183.  — The  Pediment,  above  the  entablature,  is  the  triangu¬ 
lar  portion  which  is  formed  by  the  inclined  edges  of  the  roof  at 
the  end  of  the  building.  In  Gothic  architecture,  the  pediment  is 
called,  a  gable. 

184.  — The  Tympanum  is  the  perpendicular  triangular  surface 
which  is  enclosed  by  the  cornice  of  the  pediment. 

185.  — The  Attic  is  a  small  order,  consisting  of  pilasters 
and  entablature,  raised  above  a  larger  order,  instead  of  a  pedi¬ 
ment.  An  attic  story  is  the  upper  story,  its  windows  being  usually 
square. 


10 


74 


AMERICAN  HOUSE-CARPENTER. 


186.  — An  order,  in  architecture,  has  its  several  parts  and  mem¬ 
bers  proportioned  to  one  another  by  a  scale  of  60  equal  parts, 
which  are  called  minutes.  If  the  height  of  buildings  were  al¬ 
ways  the  same,  the  scale  of  equal  parts  would  be  a  fixed  quan¬ 
tity — an  exact  number  of  feet  and  inches.  But  as  buildings  are 
erected  of  different  heights,  the  column  and  its  accompaniments 
are  required  to  be  of  different  dimensions.  To  ascertain  the  scale 
of  equal  parts,  it  is  necessary  to  know  the  height  to  which  the 
whole  order  is  to  be  erected.  This  must  be  divided  by  the  num¬ 
ber  of  diameters  which  is  directed  for  the  order  under  considera¬ 
tion.  Then  the  quotient  obtained  by  such  division,  is  the  length 
of  the  scale  of  equal  parts — and  is,  also,  the  diameter  of  the 
column  next  above  the  base.  For  instance,  in  the  Grecian  Doric 
order  the  whole  height,  including  column  and  entablature,  is  8 
diameters.  Suppose  now  it  were  desirable  to  construct  an  exam¬ 
ple  of  this  order,  forty  feet  high.  Then  40  feet  divided  by  8, 
gives  5  feet  for  the  length  of  the  scale  ;  and  this  being  divided  by 
60,  the  scale  is  completed.  The  upright  columns  of  figures, 
marked  H and  P,  by  the  side  of  the  drawings  illustrating  the  orders, 
designate  the  height  and  the  projection  of  the  members.  The 
projection  of  each  member  is  reckoned  from  a  line  passing  through 
the  axis  of  the  column,  and  extending  above  it  to  the  top  of  the 
entablature.  The  figures  represent  minutes,  or  60ths,  of  the 
major  diameter  of  the  shaft  of  the  column. 

187.  — Grecian  Styles.  The  original  method  of  building 
among  the  Greeks,  was  in  what  is  called  the  Doric  order :  to 
this  were  afterwards  added  the  Ionic  and  the  Corinthian. 
These  three  were  the  only  styles  known  among  them.  Each 
is  distinguished  from  the  other  two,  by  not  only  a  peculiarity 
of  some  one  or  more  of  its  principal  parts,  but  also  by  a 
particular  destination.  The  character  of  the  Doric  is  robust, 
manly  and  Herculean-like ;  that  of  the  Ionic  is  more  delicate, 
feminine,  matronly ;  while  that  of  the  Corinthian  is  extremely 
delicate,  youthful  and  virgin-like.  However  they  may  differ  in 


ARCHITECTURE. 


75 


their  general  character,  they  are  alike  famous  for  grace  and  dig¬ 
nity,  elegance  and  grandeur,  to  a  high  degree  of  perfection. 

188.  — The  Doric  Order  is  so  ancient  that  its  origin  is  un- 
Known — although  some  have  pretended  to  have  discovered  it. 
But  the  most  general  opinion  is,  that  it  is  an  improvement  upon 
the  original  log  huts  of  the  Grecians.  These  no  doubt  were  very 
rude,  and  perhaps  not  unlike  the  following  figure. 

The  trunks  of  trees,  set 
perpendicularly  to  support 
the  roof,  may  be  taken  for 
columns  ;  the  tree  laid  upon 
the  tops  of  the  perpendicu¬ 
lar  ones,  the  architrave  ;  the 
ends  of  the  cross-beams 
which  rest  upon  the  architrave,  the  triglyphs ;  the  tree  laid  on 
the  cross-beams  as  a  support  for  the  ends  of  the  rafters,  the  bed¬ 
moulding  of  the  cornice  ;  the  ends  of  the  rafters  which  project 
beyond  the  bed-moulding,  the  mutules;  and  perhaps  the  projection 
of  the  roof  in  front,  to  screen  the  entrance  from  the  weather,  gave 
origin  to  the  portico. 

The  peculiarities  of  the  Doric  order  are  the  triglyphs — those 
parts  of  the  frieze  which  have  perpendicular  channels  cut  in  their 
surface;  the  absence  of  abase  to  the  column — as  also  of  fillets 
between  the  flutings  of  the  column,  and  the  plainness  of  the 
oapital.  The  triglyphs  are  to  be  so  disposed  that  the  width  of 
the  metopes — the  spaces  between  the  triglyphs — shall  be  equal  to 
their  height. 

189.  — The  inter columniation,  or  space  between  the  columns, 
is  regulated  by  placing  the  centres  of  the  columns  under  the  cen¬ 
tres  of  the  triglyphs — except  at  the  angle  of  the  building ;  where, 
as  may  be  seen  in  Fig.  110,  one  edge  of  the  triglyph  must  be 
over  the  centre  of  the  column.  Where  the  columns  are  so  dis¬ 
posed  that  one  of  them  stands  beneath  every  other  triglyph,  the 
arrangement  is  called,  mono-triglyph ,  and  is  most  common. 


76 


DORIC  ORDER, 


ARCHITECTURE. 


77 


When  a  column  is  placed  beneath  every  third  triglyph,  the  ar¬ 
rangement  is  called  diastyle ;  and  when  beneath  every  fourth, 
arceostyle.  This  last  style  is  the  worst,  and  is  seldom  practised. 

190.  — The  Doric  order  is  suitable  for  buildings  that  are  des¬ 
tined  for  national  purposes,  for  banking-houses,  &c.  Its  appear¬ 
ance,  though  massive  and  grand,  is  nevertheless  rich  and  grace¬ 
ful.  The  Custom-House  and  the  Union  Bank,  in  New- York  city, 
are  good  specimens  of  this  order. 

191.  — The  Ionic  Order.  The  Doric  was  for  some  time  the 
only  order  in  use  among  the  Greeks.  They  gave  their  attention 
to  the  cultivation  of  it,  until  perfection  seems  to  have  been  at¬ 
tained.  Their  temples  were  the  principal  objects  upon  which 
their  skill  in  the  art  was  displayed ;  and  as  the  Doric  order  seems 
to  have  been  well  fitted,  by  its  massive  proportions,  to  represent 
the  character  of  their  male  deities  rather  than  the  female,  there 
seems  to  have  been  a  necessity  for  another  style  which  should  be 
emblematical  of  feminine  graces,  and  with  which  they  might 
decorate  such  temples  as  were  dedicated  to  the  goddesses.  Hence 
the  origin  of  the  Ionic  order.  This  was  invented,  according  to 
historians,  by  Hermogenes  of  Alabanda ;  and  he  being  a  native 
of  Caria,  then  in  the  possession  of  the  Ionians,  the  order  was 
called,  the  Ionic. 

192.  — The  distinguishing  features  of  this  order  are  the  volutes , 
or  spirals  of  the  capital ;  and  the  dentils  among  the  bed-mould¬ 
ings  of  the  cornice  :  although  in  some  instances,  dentils  are  want¬ 
ing.  The  volutes  are  said  to  have  been  designed  as  a  represen¬ 
tation  of  curls  of  hair  on  the  head  of  a  matron,  of  whom  the 
whole  column  is  taken  as  a  semblance. 

193.  — The  intercolumn iation  of  this  and  the  other  orders — 
both  Roman  and  Grecian,  with  the  exception  of  the  Doric — are 
distinguished  as  follows.  When  the  interval  is  one  and  a  half 
diameters,  it  is  called,  pycnostyle ,  or  columns  thick-set ;  when 
two  diameters,  systyle ;  when  two  and  a  quarter  diameters, 
eustyle  ;  when  three  diameters,  diastyle ;  and  when  more  than 


7  dia.  33m 


78 


IONIC. 


Fig.  11L 


ARCHITECTURE. 


79 


three  diameters,  arcBostyle ,  or  columns  thin-set.  In  all  the  orders, 
when  there  are  four  columns  in  one  row,  the  arrangement  is 
called,  tetrastyle  ;  when  there  are  six  in  a  row,  hexastyle  ;  and 
when  eight,  octastyle. 

194. — The  Ionic  order  is  appropriate  for  churches,  colleges, 
seminaries,  libraries,  all  edifices  dedicated  to  literature  and  the 
arts,  and  all  places  of  peace  and  tranquillity.  The  front  of  the 
Merchants’  Exchange,  New- York  city,  is  a  good  specimen  of  this 
order. 


195. —  To  describe  the  Ionic  volute.  Draw  a  perpendicular 
from  a  to  s,  {Fig.  112,)  and  make  a  s  equal  to  20  min.  or  to  f  of 
the  whole  height,  a  c  ;  draw  5  o,  at  right  angles  to  5  a,  and  equal 
to  14  min. ;  upon  o,  with  2£  min.  for  radius,  describe  the  eye  of 
the  volute ;  about  o,  the  centre  of  the  eye,  draw  the  square,  r  1 1 
2,  with  sides  equal  to  half  the  diameter  of  the  eye,  viz.,  2]  min., 
and  divide  it  into  144  equal  parts,  as  shown  at  Fig.  113.  The 
several  centres  in  rotation  are  at  the  angles  formed  by  the  heavy 
lines,  as  figured,  1, 2,  3,  4,  5,  6,  &c.  The  position  of  these  an¬ 
gles  is  determined  by  commencing  at  the  point,  1,  and  making 
each  heavy  line  one  part  less  in  length  than  the  preceding  one. 
No.  1  is  the  centre  for  the  arc,  a  b,  {Fig.  112  ;)  2  is  the  centre  for 
the  arc,  be;  and  so  on  to  the  last.  The  inside  spiral  line  is  to  be 
described  from  the  centres,  x,  x,  x,  &c.,  {Fig.  113,)  being  the 
centre  of  the  first  small  square  towards  the  middle  of  the  eye 
from  the  centre  for  the  outside  arc.  The  breadth  of  the  fillet  at 
a  j,  is  to  be  made  equal  to  2-,a0-  min.  This  is  for  a  spiral  of  three 
revolutions ;  but  one  of  any  number  of  revolutions,  as  4  or  6, 


ARCHITECTURE. 


81 


may  be  drawn,  by  dividing  o  /,  {Fig.  113,)  into  a  corresponding 
number  of  equal  parts.  Then  divide  the  part  nearest  the  centre, 
o,  into  two  parts,  as  at  h  ;  join  o  and  1,  also  o  and  2  ;  draw  h  3,  pa¬ 
rallel  to  o  1,  and  h  4,  parallel  to  o  2 ;  then  the  lines,  o  1,  o  2,  h  3,  h 
4,  will  determine  the  length  of  the  heavy  lines,  and  the  place  of 
the  centres.  (See  Art.  396.) 

196.  — The  Corinthian  Order  is  in  general  like  the  Ionic, 
though  the  proportions  are  lighter.  The  Corinthian  displays  a 
more  airy  elegance,  a  richer  appearance ;  but  its  distinguishing 
feature  is  its  beautiful  capital.  This  is  generally  supposed  to  have 
had  its  origin  in  the  capitals  of  the  columns  of  Egyptian  temples  ; 
which,  though  not  approaching  it  in  elegance,  have  yet  a  similari¬ 
ty  of  form  with  the  Corinthian.  The  oft-repeated  story  of  its 
origin  which  is  told  by  Vitruvius — an  architect  who  flourished  in 
Rome,  in  the  days  of  Augustus  Ceesar — though  pretty  generally 
considered  to  be  fabulous,  is  nevertheless  worthy  of  being  again 
recited.  It  is  this :  a  young  lady  of  Corinth  was  sick,  and 
finally  died.  Her  nurse  gathered  into  a  deep  basket,  such  trinkets 
and  keepsakes  as  the  lady  had  been  fond  of  when  alive,  and 
placed  them  upon  her  grave  ;  covering  the  basket  with  a  flat  stone 
or  tile,  that  its  contents  might  not  be  disturbed.  The  basket  was 
placed  accidentally  upon  the  stem  of  an  acanthus  plant,  which, 
shooting  forth,  enclosed  the  basket  with  its  foliage  ;  some  of  which, 
reaching  the  tile,  turned  gracefully  over  in  the  form  of  a  volute. 

A  celebrated  sculptor,  Calima- 
chus,  saw  the  basket  thus  decorated, 
and  from  the  hint  which  it  sug¬ 
gested,  conceived  and  constructed  a 
capital  for  a  column.  This  was 
called  Corinthian  from  the  fact  that  it 
was  invented  and  first  made  use  of 
Fig- 1H-  at  Corinth. 

197.  — The  Corinthian  being  the  gayest,  the  richest  and  most 
lovely  of  all  the  orders,  it  is  appropriate  for  edifices  which  are 

11 


CORINTHIAN. 


82 


ARCHITECTURE. 


83 


dedicated  to  amusement,  banqueting  and  festivity — for  all  places 
where  delicacy,  gayety  and  splendour  are  desirable. 

198.  — In  addition  to  the  three  regular  orders  of  architecture,  it 
was  sometimes  customary  among  the  Greeks — and  afterwards 
among  other  nations — to  employ  representations  of  the  human 
form,  instead  of  columns,  to  support  entablatures  ;  these  were 
called  Persians  and  Caryatides. 

199.  — Persians  are  statues  of  men,  and  are  so  called  in  com¬ 
memoration  of  a  victory  gained  over  the  Persians  by  Pausanias. 
The  Persian  prisoners  were  brought  to  Athens  and  condemned  to 
abject  slavery ;  and  in  order  to  represent  them  in  the  lowest  state 
of  servitude  and  degradation,  the  statues  were  loaded  with  the 
heaviest  entablature,  the  Doric. 

200.  — Caryatides  are  statues  of  women  dressed  in  long  robes 
after  the  Asiatic  manner.  Their  origin  is  as  follows.  In  a  war 
between  the  Greeks  and  the  Caryans,  the  latter  were  totally  van¬ 
quished,  their  male  population  extinguished,  and  their  females 
carried  to  Athens.  To  perpetuate  the  memory  of  this  event, 
statues  of  females,  having  the  form  and  dress  of  the  Caryans,  were 
erected,  and  crowned  with  the  Ionic  or  Corinthian  entablature. 
The  caryatides  were  generally  formed  of  about  the  human  size, 
but  the  persians  much  larger  ;  in  order  to  produce  the  greater  awe 
and  astonishment  in  the  beholder.  The  entablatures  were  pro¬ 
portioned  to  a  statue  in  like  manner  as  to  a  column  of  the  same 
height. 

201.  — These  semblances  of  slavery  have  been  in  frequent  use 
among  moderns  as  well  as  ancients ;  and  as  a  relief  from  the 
stateliness  and  formality  of  the  regular  orders,  are  capable  of 
forming  a  thousand  varieties  ;  yet  in  a  land  of  liberty  such  marks 
•of  human  degradation  ought  not  to  be  perpetuated. 

202.  — Roman  Styles.  Strictly  speaking,  Rome  had  no 
architecture  of  her  own — all  she  possessed  was  borrowed  from 
■other  nations.  Before  the  Romans  exchanged  intercourse  with 
the  Greeks,  they  possessed  some  edifices  of  considerable  extent 


84 


AMERICAN  HOUSE-CARPENTER. 


and  merit,  which  were  erected  by  architects  from  Etruria ;  but 
Rome  was  principally  indebted  to  Greece  for  what  she  acquired 
of  the  art.  Although  there  is  no  such  thing  as  an  architecture  of 
Roman  invention,  yet  no  nation,  perhaps,  ever  was  so  devoted  to 
the  cultivation  of  the  art  as  the  Roman.  Whether  we  consider 
the  number  and  extent  of  their  structures,  or  the  lavish  richness 
and  splendour  with  which  they  were  adorned,  we  are  compelled 
to  yield  to  them  our  admiration  and  praise.  At  one  time,  under 
the  consuls  and  emperors,  Rome  employed  400  architects.  The 
public  works — such  as  theatres,  circuses,  baths,  aqueducts,  &c.— 
were,  in  extent  and  grandeur,  beyond  any  thing  attempted  in 
modern  times.  Aqueducts  were  built  to  convey  water  from  a 
distance  of  60  miles  or  more.  In  the  prosecution  of  this  work, 
rocks  and  mountains  were  tunnelled,  and  valleys  bridged.  Some 
of  the  latter  descended  200  feet  below  the  level  of  the  water ;  and 
in  passing  them  the  canals  were  supported  by  an  arcade,  or  suc¬ 
cession  of  arches.  Public  baths  are  spoken  of  as  large  as  cities ; 
being  fitted  up  with  numerous  conveniences  for  exercise  and 
amusement.  Their  decorations  were  most  splendid ;  indeed,  the 
exuberance  of  the  ornaments  alone  was  offensive  to  good  taste. 
So  overloaded  with  enrichments  were  the  baths  of  Diocletian, 
that  on  an  occasion  of  public  festivity,  great  quantities  of  sculp¬ 
ture  fell  from  the  ceilings  and  entablatures,  killing  many  of  the 
people. 

203. — The  three  orders  of  Greece  were  introduced  into  Rome 
in  all  the  richness  and  elegance  of  their  perfection.  But  the  luxu¬ 
rious  Romans,  not  satisfied  with  the  simple  elegance  of  their  re¬ 
fined  proportions,  sought  to  improve  upon  them  by  lavish  displays 
of  ornament.  They  transformed  in  many  instances,  the  true  ele¬ 
gance  of  the  Grecian  art  into  a  gaudy  splendour,  better  suited  to 
their  less  refined  taste.  The  Romans  remodelled  each  of  the 
orders  :  the  Doric  was  modified  by  increasing  the  height  of  the 
column  to  8  diameters  ;  by  changing  the  echinus  of  the  capital 
for  an  ovolo,  or  quarter-round,  and  adding  an  astragal  and  neck 


V 

J 


ARCHITECTURE.  85 

below  it ;  by  placing  the  centre  of  the  first  triglyph,  instead  of 
one  edge,  over  the  centre  of  the  column  ;  and  introducing  hori¬ 
zontal  instead  of  inclined  mutules  in  the  cornice.  The  Ionic 
was  modified  by  diminishing  the  size  of  the  volutes,  and,  in  some 
specimens,  introducing  a  new  capital  in  which  the  volutes  were 
diagonally  arranged.  This  new  capital  has  been  termed  modern 
Ionic.  The  favorite  order  at  Rome  and  her  colonies  was  the  Co¬ 
rinthian.  The  Roman  artists,  in  their  search  for  novelty,  sub¬ 
jected  it  to  many  alterations — especially  in  the  foliage  of  its  capi¬ 
tal.  Into  the  upper  part  of  this,  they  introduced  the  modified 
Ionic  capital ;  thus  combining  the  two  in  one.  This  change  was 
dignified  with  the  importance  of  an  order ,  and  received  the  ap¬ 
pellation  Composite,  or  Roman:  the  best  specimen  of  which  is 
found  in  the  Arch  of  Titus.  This  style  was  not  much  used 
among  the  Romans  themselves,  and  is  but  slightly  appreciated 
now.  Its  decorations  are  too  profuse — a  standing  monument  of 
the  luxury  of  the  age  in  which  it  was  invented. 

204.  — The  Tuscan  Order  is  said  to  have  been  introduced 
to  the  Romans  by  the  Etruscan  architects,  and  to  have  been 
the  only  style  used  in  Italy  before  the  introduction  of  the 
Grecian  orders.  However  this  may  be,  its  similarity  to  the 
Doric  order  gives  strong  indications  of  its  having  been  a 
rude  imitation  of  that  style :  this  is  very  probable,  since  his¬ 
tory  informs  us  that  the  Etruscans  held  intercourse  with  the 
Greeks  at  a  remote  period.  The  rudeness  of  this  order  prevented 
its  extensive  use  in  Italy.  All  that  is  known  concerning  it  is  from 
Yitruvius — no  remains  of  buildings  in  this  style  being  found 
among  ancient  ruins. 

205.  For  mills,  factories,  markets,  barns,  stables,  &c.,  where 
utility  and  strength  are  of  more  importance  than  beauty,  the  im¬ 
proved  modification  of  this  order,  called  the  modern  Tuscan, 
{Fig.  116,)  will  be  useful ;  and  its  simplicity  recommends  it 
where  economy  is  desirable. 

206.  — Egyptian  Style.  The  architecture  of  the  ancient 


86 


TUSCAN. 


ARCHITECTURE. 


87 


Egyptians — to  which  that  of  the  ancient  Hindoos  bears  some  re¬ 
semblance — is  characterized  by  boldness  of  outline,  solidity  and 
grandeur.  The  amazing  labyrinths  and  extensive  artificial  lakes, 
the  splendid  palaces  and  gloomy  cemeteries,  the  gigantic  pyramids 
and  towering  obelisks,  of  the  Egyptians,  were  works  of  immen¬ 
sity  and  durability ;  and  their  extensive  remains  are  enduring 
proofs  of  the  enlightened  skill  of  this  once-powerful,  but  long  since 
extinct  nation.  The  principal  features  of  the  Egyptian  Style  of 
architecture  are — uniformity  of  plan,  never  deviating  from  right 
lines  and  angles  ;  thick  walls,  having  the  outer  surface  slightly 
deviating  inwardly  from  the  perpendicular ;  the  whole  building 
low ;  roof  flat,  composed  of  stones  reaching  in  one  piece  from  pier 
to  pier,  these  being  supported  by  enormous  columns,  very  short  in 
proportion  to  their  height ;  the  shaft  sometimes  polygonal,  having 
no  base  but  with  a  great  variety  of  handsome  capitals,  the  foliage 
of  these  being  of  the  palm,  lotus  and  other  leaves  ;  entablatures 
having  simply  an  architrave,  crowned  with  a  huge  cavetto  orna¬ 
mented  with  sculpture  ;  and  the  intercolumniation  very  narrow, 
usually  li  diameters  and  seldom  exceeding  2£.  In  the  remains 
of  a  temple,  the  walls  were  found  to  be  24  feet  thick  ;  and  at  the 
gates  of  Thebes,  the  walls  at  the  foundation  were  50  feet  thick 
and  perfectly  solid.  The  immense  stones  of  which  these,  as  well 
as  Egyptian  walls  generally,  were  built,  had  both  their  inside  and 
outside  surfaces  faced,  and  the  joints  throughout  the  body  of  the 
wall  as  perfectly  close  as  upon  the  outer  surface.  For  this  reason, 
as  well  as  that  the  buildings  generally  partake  of  the  pyramidal 
form,  arise  their  great  solidity  and  durability.  The  dimensions 
and  extent  of  the  buildings  may  be  judged  from  the  temple  of 
Jupiter  at  Thebes,  which  was  1400  feet  long  and  300  feet  wide — 
exclusive  of  the  porticos,  of  which  there  was  a  great  number. 

It  is  estimated  by  Mr.  Gliddon,  U.  S.  consul  in  Egypt,  that  not 
less  than  25,000,000  tons  of  hewn  stone  were  employed  in  the 
erection  of  the  Pyramids  of  Memphis  alone, — or  enough  to  con¬ 
struct  3,000  Bunker-Hill  monuments.  Some  of  the  blocks  are  40 


88 


EGYPTIAN. 


H.  P. 


# 


ARCHITECTURE. 


89 


feet  long,  and  polished  with  emery  to  a  surprising  degree.  It  is 
conjectured  that  the  stone  for  these  pyramids  was  brought,  by 
rafts  and  canals,  from  a  distance  of  6  or  7  hundred  miles. 

207.  — The  general  appearance  of  the  Egyptian  style  of  archi¬ 
tecture  is  that  of  solemn  grandeur — amounting  sometimes  to 
sepulchral  gloom.  For  this  reason  it  is  appropriate  for  cemete¬ 
ries,  prisons,  &c. ;  and  being  adopted  for  these  purposes,  it  is 
gradually  gaining  favour. 

A  great  dissimilarity  exists  in  the  proportion;  form  and  general 
features  of  Egyptian  columns.  In  some  instances,  there  is  no 
uniformity  even  in  those  of  the  same  building,  each  differing 
from  the  others  either  in  its  shaft  or  capital.  For  practical  use 
in  this  country,  Fig.  117  may  be  taken  as  a  standard  of  this 
style.  The  Halls  of  Justice  in  Centre-street,  New- York  city,  is 
a  building  in  general  accordance  with  the  principles  of  Egyptian 
architecture. 

Buildings  in  General. 

208. — That  style  of  architecture  is  to  be  preferred  in  which 
utility,  stability  and  regularity,  are  gracefully  blended  with  gran¬ 
deur  and  elegance.  But  as  an  arrangement  designed  for  a  warm 
country  would  be  inappropriate  for  a  colder  climate,  it  would  seem 
that  the  style  of  building  ought  to  be  modified  to  suit  the  wants 
of  the  people  for  whom  it  is  designed.  High  roofs  to  resist  the 
pressure  of  heavy  snows,  and  arrangements  for  artificial  heat,  are 
indispensable  in  northern  climes  ;  while  they  would  be  regarded 
as  entirely  out  of  place  in  buildings  at  the  equator. 

209.  — Among  the  Greeks,  architecture  was  employed  chiefly 
upon  their  temples  and  other  large  buildings  ;  and  the  proportions 
of  the  orders,  as  determined  by  them,  when  executed  to  such 
large  dimensions,  have  the  happiest  effect.  But  when  used  for 
small  buildings, porticos,  porches,  &c.,  especially  in  country-places, 
they  are  rather  heavy  and  clumsy ;  in  such  cases,  more  slender 
proportions  will  be  found  to  produce  a  better  effect.  The 

12 


90 


AMERICAN  HOUSE-CARPENTER. 


English  cottage-style  is  rather  more  appropriate,  and  is  becom¬ 
ing  extensively  practised  for  small  buildings  in  the  country. 

210.  — Every  building  should  bear  an  expression  suited  to  its 
destination.  If  it  be  intended  for  national  purposes,  it  should  be 
magnificent — grand ;  for  a  private  residence,  neat  and  modest ; 
for  a  banqueting-house,  gay  and  splendid ;  for  a  monument  or 
cemetery,  gloomy — melancholy  ;  or,  if  for  a  church,  majestic  and 
graceful.  By  some  it  has  been  said — “somewhat  dark  and 
gloomy,  as  being  favourable  to  a  devotional  state  of  feeling but 
such  impressions  can  only  result  from  a  misapprehension  of  the 
nature  of  true  devotion.  “  Her  ways  are  ways  of  pleasantness , 
and  all  her  paths  are  peace.”  The  church  should  rather  be  a  type 
of  that  brighter  world  to  which  it  leads. 

211.  — However  happily  the  several  parts  of  an  edifice  may  be 
disposed,  and  however  pleasing  it  may  appear  as  a  whole,  yet 
much  depends  upon  its  site ,  as  also  upon  the  character  and  style 
of  the  structures  in  its  immediate  vicinity,  and  the  degree  of  cul¬ 
tivation  of  the  adjacent  country.  A  splendid  country-seat  should 
have  the  out-houses  and  fences  in  the  same  style  with  itself,  the 
trees  and  shrubbery  neatly  trimmed,  and  the  grounds  well  cul¬ 
tivated. 

212.  — Europeans  express  surprise  that  so  many  houses  in  this 
country  are  built  of  wood.  And  yet,  in  a  new  country,  where 
wood  is  plenty,  that  this  should  be  so  is  no  cause  for  wonder. 
Still,  the  practice  should  not  be  encouraged.  Buildings  erected 
with  brick  or  stone  are  far  preferable  to  those  of  wood  ;  they  are 
more  durable  ;  not  so  liable  to  injury  by  fire,  nor  to  need  repairs  ; 
and  will  be  found  in  the  end  quite  as  economical.  A  wooden 
house  is  suitable  for  a  temporary  residence  only ;  and  those  who 
would  bequeath  a  dwelling  to  their  children,  will  endeavour  to 
build  with  a  more  durable  material.  Wooden  cornices  and  gut¬ 
ters,  attached  to  brick  houses,  are  objectionable — not  only  on  ac¬ 
count  of  their  frail  nature,  but  also  because  they  render  the  build¬ 
ing  liable  to  destruction  by  fire. 


91 


Fig.  118 


Fig.  119. 


92 


AMERICAN  HOUSE-CARPENTER. 


213. — Dwelling  houses  are  built  of  various  dimensions  and 
styles,  according  to  their  destination ;  and  to  give  designs  and  di¬ 
rections  for  their  erection,  it  is  necessary  to  know  their  situation 
and  object.  A  dwelling  intended  for  a  gardener,  would  require 
very  different  dimensions  and  arrangements  from  one  intended  for 
a  retired  gentlemen — with  his  servants,  horses,  &c. ;  nor  would 
a  house  designed  for  the  city,  be  appropriate  for  the  country.  For 
city  houses,  arrangements  that  would  be  convenient  for  one  fa¬ 
mily,  might  be  very  inconvenient  for  two  or  more.  Fig.  118,  119, 
120  and  121,  represent  the  ichno graphical  projection ,  or  ground- 
plan,  of  the  floors  of  an  ordinary  city  house,  designed  to  be  occupied 
by  one  family  only.  Fig.  122  is  an  elevation,  or  front-view,  of 
the  same  house :  all  these  plans  are  drawn  at  the  same  scale — 
which  is  that  at  the  bottom  of  Fig.  122. 

Fig.  118  is  a  plan  of  the  basement. 
a  is  the  dining-room. 
b — kitchen. 
c — wash-room. 
cl,  d,  d, — wash-troughs. 
e,  e, — pantries  with  shelving. 

/ — passage  having  shelves,  drawers,  &c.,  on  one  side,  and 
clothes-hooks  on  the  other. 
g — kitchen-dresser. 
h,  i, — front  and  rear  areas. 

Fig,  119 — plan  of  the  first-story. 
j,j  — parlours. 
k — library. 

I — portico. 

Fig.  120 — plan  of  the  second-story. 
a — toilet  and  sitting  room. 
b — principal  bed-chamber, 
c — bath-room. 
d,  d, — bed-chambers. 

e— passage  with  wardrobe  and  clothes-hooks. 


93 


Fig.  120. 


Fig.  121. 


94 


AMERICAN  HOUSE-CARPENTER. 


Fig.  121 — plan  of  the  attic-story. 

/—nursery, 

g, g,  g,  g i — bed-chambers, 

h,  h,  h,  h,  h, — wardrobes, 

i — pantry  with  shelves, 

j — step-ladder  leading  to  roof. 

Fig.  122 — front  elevation. 

a — section, 

b — front, 

These  are  introduced  to  give  some  general  ideas  of  the  princi¬ 
ples  to  be  followed  in  designing  city  houses.  The  width  of  city 
lots  is  ordinarily  25  feet,  but  as  it  has  become  a  common  practice 
to  reduce  this  size,  on  account  of  the  enhanced  value  of  land,  the 
plans  here  given  are  designed  for  a  lot  only  20  feet  wide — the  or¬ 
dinary  width  of  many  buildings  of  this  class.  In  placing  the 
chimneys,  make  the  parlours  of  equal  size,  and  set  the  chimney- 
breast  in  the  middle  of  the  space  between  the  sliding-door  parti¬ 
tion  and  the  front  (and  rear)  walls.  The  basement  chimney- 
breasts  may  be  placed  in  the  middle  of  the  side  of  the  room,  as 

there  is  but  one  flue  to  pass  through  the  chimney-breast  above ; 

. 

but  in  the  second-story,  as  there  is  two  flues,  one  from  the  base¬ 
ment  and  one  from  the  parlour,  the  breast  will  have  to  be  placed 
nearly  perpendicular  over  the  parlour  breast,  so  as  to  receive  the 
flues  within  the  jambs  of  the  fire-place.  As  it  is  desirable  to 
have  the  chimney-breast  as  near  the  middle  of  the  room  as  pos¬ 
sible,  it  may  be  placed  a  few  inches  towards  that  point  from  over 
the  breast  below.  So  in  arranging  those  of  the  stories  above, 
always  make  provision  for  the  flues  from  below. 

214. — In  placing  the  stairs,  there  should  be  at  least  as  much 
room  in  the  passage  at  the  side  of  the  stairs,  as  upon  them ;  and  in 
regard  to  the  length  of  the  passage  in  the  second  story,  there  must 
be  room  for  the  doors  which  open  from  each  of  the  principal  rooms 
into  the  hall,  and  more  if  the  stairs  require  it.  Having  assigned 
a  position  for  the  stairs  of  the  second  story,  let  the  winders  of 


v 


95 


i 


96 


AMERICAN  HOUSE-CARPENTER. 


the  other  stories  be  placed  perpendicularly  over  and  under  them ; 
and  be  careful  to  provide  for  head-room.  To  ascertain  this,  when 
it  is  doubtful,  it  is  well  to  draw  a  vertical  section  of  the  whole 
stairs  ;  but  in  ordinary  cases,  this  is  not  necessary.  To  dispose 
the  windows  properly,  the  middle  window  of  each  story  should 
be  exactly  in  the  middle  of  the  front ;  but  the  pier  between  the 
two  windows  which  light  the  parlour,  should  be  in  the  centre  of 
that  room  ;  because  when  chandeliers  or  any  similar  ornaments, 
hang  from  the  centre-pieces  of  the  parlour  ceilings,  it  is  important, 
in  order  to  give  the  better  effect,  that  the  pier-glasses  at  the  front 
and  rear,  be  in  a  range  with  them.  If  both  these  objects  cannot 
be  attained,  an  approximation  to  each  must  be  attempted.  The 
piers  should  in  no  case  be  less  in  width  than  the  window  open¬ 
ings,  else  the  blinds  or  shutters  when  thrown  open  will  interfere 
with  one  another ;  in  general  practice,  it  is  well  to  make  the  out¬ 
side  piers  f  of  the  width  of  one  of  the  middle  piers.  When  this 
is  desirable,  deduct  the  amount  of  the  three  openings  from  the 
width  of  the  front,  and  the  remainder  will  be  the  amount  of  the 
width  of  all  the  piers  ;  divide  this  by  10,  and  the  product  will  be 
s  of  a  middle  pier ;  and  then,  if  the  parlour  arrangements  do  not 
interfere,  give  twice  this  amount  to  each  corner  pier,  and  three 
times  the  same  amount  to  each  of  the  middle  piers. 

PRINCIPLES  OF  ARCHITECTURE. 

215. — In  the  construction  of  the  first  habitations  of  men,  frail 
and  rude  as  they  must  have  been,  the  first  and  principal  object 
was,  doubtless,  utility — a  mere  shelter  from  sun  and  rain.  But 
as  successive  storms  shattered  the  poor  tenement,  man  was  taught 
by  experience  the  necessity  of  building  with  an  idea  to  durability. 
And  when  in  his  walks  abroad,  the  symmetry,  proportion  and 
beauty  of  nature  met  his  admiring  gaze,  contrasting  so  strangely 
with  the  misshapen  and  disproportioned  work  of  his  own  hands, 
he  was  led  to  make  gradual  changes  ;  till  his  abode  was  rendered 


ARCHITECTURE. 


97 


not  only  commodious  and  durable,  but  pleasant  in  its  appearance ; 
and  building  became  a  fine-art,  having  utility  for  its  basis. 

216.  — In  all  designs  for  buildings  of  importance,  utility,  dura¬ 
bility  and  beauty,  the  first  great  principles  of  architecture,  should 
be  pre-eminent.  In  order  that  the  edifice  be  useful,  commodious 
and  comfortable,  the  arrangement  of  the  apartments  should  be 
such  as  to  fit  them  for  their  several  destinations  ;  for  public  as¬ 
semblies,  oratory,  state,  visitors,  retiring,  eating,  reading,  sleeping, 
bathing,  dressing,  &c. — these  should  each  have  its  own  peculiar 
form  and  situation.  To  accomplish  this,  and  at  the  same  time  to 
make  their  relative  situation  agreeable  and  pleasant,  producing 
regularity  and  harmony,  require  in  some  instances  much  skill  and 
sound  judgment.  Convenience  and  regularity  are  very  import¬ 
ant,  and  each  should  have  due  attention  ;  yet  when  both  cannot 
be  obtained,  the  latter  should  in  most  cases  give  place  to  the  for¬ 
mer.  A  building  that  is  neither  convenient  nor  regular,  whatever 
other  good  qualities  it  may  possess,  will  be  sure  of  disappro¬ 
bation. 

217.  — The  utmost  importance  should  be  attached  to  such  ar¬ 
rangements  as  are  calculated  to  promote  health  :  among  these,  ven¬ 
tilation  is  by  no  means  the  least.  For  this  purpose,  the  ceilings  of 
the  apartments  should  have  a  respectable  height ;  and  the  sky¬ 
light,  or  any  part  of  the  roof  that  can  be  made  moveable,  should 
be  arranged  with  cord  and  pullies,  so  as  to  be  easily  raised  and 
lowered.  Small  openings  near  the  ceiling,  that  may  be  closed  at 
pleasure,  should  be  made  in  the  partitions  that  separate  the  rooms 
from  the  passages — especially  for  those  rooms  which  are  used  for 
sleeping  apartments.  All  the  apartments  should  be  so  arranged 
as  to  secure  their  being  easily  kept  dry  and  clean.  In  dwellings, 
suitable  apartments  should  be  fitted  up  for  bathing ,  with  all  the 
necessaty  apparatus  for  conveying  the  water. 

218.  — To  insure  stability  in  an  edifice,  it  should  be  designed 
upon  well-known  geometrical  principles :  such  as  science  has  de¬ 
monstrated  to  be  necessary  and  sufficient  for  firmness  and  dura- 

13 


98 


AMERICAN  HOUSE-CARPENTER. 


bility.  It  is  well,  also,  that  it  have  the  appearance  of  stability  as 
well  as  the  reality  ;  for  should  it  seem  tottering  and  unsafe,  the 
sensation  of  fear,  rather  than  those  of  admiration  and  pleasure, 
will  be  excited  in  the  beholder.  To  secure  certainty  and  accu¬ 
racy  in  the  application  of  those  principles,  a  knowledge  of  the 
strength  and  other  properties  of  the  materials  used,  is  indispensa¬ 
ble  ;  and  in  order  that  the  whole  design  be  so  made  as  to  be 
capable  of  execution,  a  practical  knowledge  of  the  requisite 
mechanical  operations  is  quite  important. 

219.  — The  elegance  of  an  architectural  design,  although  chiefly 
depending  upon  a  just  proportion  and  harmony  of  the  parts,  will 
be  promoted  by  the  introduction  of  ornaments — provided  this  be 
judiciously  performed.  For  enrichments  should  not  only  be  of  a 
proper  character  to  suit  the  style  of  the  building,  but  should  also 
have  their  true  position,  and  be  bestowed  in  proper  quantity.  The 
most  common  fault,  and  one  which  is  prominent  in  Roman  archi¬ 
tecture,  is  an  excess  of  enrichment :  an  error  which  is  carefully 
to  be  guarded  against.  But  those  who  take  the  Grecian  models 
for  their  standard,  will  not  be  liable  to  go  to  that  extreme.  In 
ornamenting  a  cornice,  or  any  other  assemblage  of  mouldings,  at 
least  every  alternate  member  should  be  left  plain ;  and  those  that 
are  near  the  eye  should  be  more  finished  than  those  which  are  dis¬ 
tant.  Although  the  characteristics  of  good  architecture  are  utili¬ 
ty  and  elegance,  in  connection  with  durability,  yet  some  buildings 
are  designed  expressly  for  use,  and  others  again  for  ornament :  in 
the  former,  utility,  and  in  the  latter,  beauty,  should  be  the  gov¬ 
erning  principle. 

220.  — The  builder  should  be  intimately  acquainted  with  the 
principles  upon  which  the  essential,  elementary  parts  of  a  build¬ 
ing  are  founded.  A  scientific  knowledge  of  these  will  insure 
certainty  and  security,  and  enable  the  mechanic  to  erect  the  most 
extensive  and  lofty  edifices  with  confidence.  The  more  important 
parts  are  the  foundation,  the  column,  the  wall,  the  lintel,  the  arch, 
the  vault,  the  dome  and  the  roof.  A  separate  description  of  the 


ARCHITECTURE. 


99 


peculiarities  of  each,  would  seem  to  be  necessary;  and  cannot 
perhaps  be  better  expressed  than  in  the  following  language  of  a 
modern  writer  on  this  subject. 

221.  — “In  laying  the  Foundation  of  any  building,  it  is  ne¬ 
cessary  to  dig  to  a  certain  depth  in  the  earth,  to  secure  a  solid 
basis,  below  the  reach  of  frost  and  common  accidents.  The 
most  solid  basis  is  rock,  or  gravel  which  has  not  been  moved. 
Next  to  these  are  clay  and  sand,  provided  no  other  excavations 
have  been  made  in  the  immediate  neighbourhood.  From  this 
basis  a  stone  wall  is  carried  up  to  the  surface  of  the  ground,  and 
constitutes  the  foundation.  Where  it  is  intended  that  the  super¬ 
structure  shall  press  unequally,  as  at  its  piers,  chimneys,  or 
columns,  it  is  sometimes  of  use  to  occupy  the  space  between  the 
points  of  pressure  by  an  inverted  arch.  This  distributes  the 
pressure  equally,  and  prevents  the  foundation  from  springing  be¬ 
tween  the  different  points.  In  loose  or  muddy  situations,  it  is 
always  unsafe  to  build,  unless  we  can  reach  the  solid  bottom 
below.  In  salt  marshes  and  flats,  this  is  done  by  depositing  tim¬ 
bers,  or  driving  wooden  piles  into  the  earth,  and  raising  walls 
upon  them.  The  preservative  quality  of  the  salt  will  keep  these 
timbers  unimpaired  for  a  great  length  of  time,  and  makes  the 
foundation  equally  secure  with  one  of  brick  or  stone. 

222.  — The  simplest  member  in  any  building,  though  by  no 
means  an  essential  one  to  all,  is  the  Column,  or  pillar.  This  is 
a  perpendicular  part,  commonly  of  equal  breadth  and  thickness, 
not  intended  for  the  purpose  of  enclosure,  but  simply  for  the  sup¬ 
port  of  some  part  of  the  superstructure.  The  principal  force 
which  a  column  has  to  resist,  is  that  of  perpendicular  pressure. 
In  its  shape,  the  shaft  of  a  column  should  not  be  exactly  cylin¬ 
drical,  but,  since  the  lower  part  must  support  the  weight  of  the 
superior  part,  in  addition  to  the  weight  which  presses  equally  on 
the  whole  column,  the  thickness  should  gradually  decrease  from 
bottom  to  top.  The  outline  of  columns  should  be  a  little  curved, 
so  as  to  represent  a  portion  of  a  very  long  spheroid,  or  paraboloid, 


100 


AMERICAN  HOUSE-CARPENTER. 


rather  than  of  a  cone.  This  figure  is  the  joint  result  of  two  cal¬ 
culations,  independent  of  beauty  of  appearance.  One  of  these 
is,  that  the  form  best  adapted  for  stability  of  base  is  that  of  a 
cone;  the  other  is,  that  the  figure,  which  would  be  of  equal 
strength  throughout  for  supporting  a  superincumbent  weight, 
would  be  generated  by  the  revolution  of  two  parabolas  round  the 
axis  of  the  column,  the  vertices  of  the  curves  being  at  its  ex¬ 
tremities.  The  swell  of  the  shafts  of  columns  was  called  the  en¬ 
tasis  by  the  ancients.  It  has  been  lately  found,  that  the  columns 
of  the  Parthenon,  at  Athens,  which  have  been  commonly  sup¬ 
posed  straight,  deviate  about  an  inch  from  a  straight  line,  and 
that  their  greatest  swell  is  at  about  one  third  of  their  height. 
Columns  in  the  antique  orders  are  usually  made  to  diminish  one 
sixth  or  one  seventh  of  their  diameter,  and  sometimes  even  one 
fourth.  The  Gothic  pillar  is  commonly  of  equal  thickness 
throughout. 

223. — The  Wall,  another  elementary  part  of  a  building,  may 
be  considered  as  the  lateral  continuation  of  the  column,  answer¬ 
ing  the  purpose  both  of  enclosure  and  support.  A  wall  must 
diminish  as  it  rises,  for  the  same  reasons,  and  in  the  same  propor¬ 
tion,  as  the  column.  It  must  diminish  still  more  rapidly  if  it  ex¬ 
tends  through  several  stories,  supporting  weights  at  different 
heights.  A  wall,  to  possess  the  greatest  strength,  must  also  con¬ 
sist  of  pieces,  the  upper  and  lower  surfaces  of  which  are  horizon¬ 
tal  and  regular,  not  rounded  nor  oblique.  The  walls  of  most  of 
the  ancient  structures  which  have  stood  to  the  present  time,  are 
constructed  in  this  manner,  and  frequently  have  their  stones  bound 
together  with  bolts  and  cramps  of  iron.  The  same  method  is 
adopted  in  such  modern  structures  as  are  intended  to  possess  great 
strength  and  durability,  and,  in  some  cases,  the  stones  are  even 
dove-tailed  together,  as  in  the  light-houses  at  Eddystone  and  Bell 
Rock.  But  many  of  our  modern  stone  walls,  for  the  sake  of 
cheapness,  have  only  one  face  of  the  stones  squared,  the  inner 
half  of  the  wall  being  completed  with  brick ;  so  that  they  can, 


ARCHITECTURE. 


101 


in  reality,  be  considered  only  as  brick  walls  faced  with  stone. 
Such  walls  are  said  to  be  liable  to  become  convex  outwardly,  from 
the  difference  in  the  shrinking  of  the  cement.  Rubble  walls  are 
made  of  rough,  irregular  stones,  laid  in  mortar.  The  stones 
should  be  broken,  if  possible,  so  as  to  produce  horizontal  surfaces. 
The  coffer  walls  of  the  ancient  Romans  were  made  by  enclosing 
successive  portions  of  the  intended  wall  in  a  box,  and  filling  it 
with  stones,  sand,  and  mortar,  promiscuously.  This  kind  of 
structure  must  have  been  extremely  insecure.  The  Pantheon, 
and  various  other  Roman  buildings,  are  surrounded  with  a  double 
brick  wall,  having  its  vacancy  filled  up  with  loose  bricks  and 
cement.  The  whole  has  gradually  consolidated  into  a  mass  of 
great  firmness. 

The  reticulated  walls  of  the  Romans,  having  bricks  with 
oblique  surfaces,  would,  at  the  present  day,  be  thought  highly 
unphilosophical.  Indeed,  they  could  not  long  have  stood,  had  it 
not  been  for  the  great  strength  of  their  cement.  Modern  brick 
walls  are  laid  with  great  precision,  and  depend  for  firmness  more 
upon  their  position  than  upon  the  strength  of  their  cement.  The 
bricks  being  laid  in  horizontal  courses,  and  continually  overlaying 
each  other,  or  breaking  joints,  the  whole  mass  is  strongly  inter¬ 
woven,  and  bound  together.  Wooden  walls,  composed  of  timbers 
covered  with  boards,  are  a  common,  but  more  perishable  kind. 
They  require  to  be  constantly  covered  with  a  coating  of  a  foreign 
substance,  as  paint  or  plaster,  to  preserve  them  from  spontaneous 
decomposition.  In  some  parts  of  France,  and  elsewhere,  a  kind 
of  wall  is  made  of  earth,  rendered  compact  by  ramming  it  in 
moulds  or  cases.  This  method  is  called  building  in  pise ,  and  is 
much  more  durable  than  the  nature  of  the  material  would  lead 
us  to  suppose.  Walls  of  all  kinds  are  greatly  strengthened  by 
angles  and  curves,  also  by  projections,  such  as  pilasters,  chimneys 
and  buttresses.  These  projections  serve  to  increase  the  breadth 
of  the  foundation,  and  are  always  to  be  made  use  of  in  large 
buildings,  and  in  walls  of  considerable  length. 


102  AMERICAN  HOUSE-CARPENTER. 

'  224. — The  Lintel,  or  beam ,  extends  in  a  right  line  over  a 
vacant  space,  from  one  column  or  wall  to  another.  The  strength 
of  the  lintel  will  be  greater  in  proportion  as  its  transverse  vertical 
diameter  exceeds  the  horizontal,  the  strength  being  always  as  the 
square  of  the  depth.  The  floor  is  the  lateral  continuation  or 
connection  of  beams  by  means  of  a  covering  of  boards. 

225. — The  Arch  is  a  transverse  member  of  a  building,  an¬ 
swering  the  same  purpose  as  the  lintel,  but  vastly  exceeding  it  in 
strength.  The  arch,  unlike  the  lintel,  may  consist  of  any  num¬ 
ber  of  constituent  pieces,  without  impairing  its  strength.  It  is, 
however,  necessary  that  all  the  pieces  should  possess  a  uniform 
shape, — the  shape  of  a  portion  of  a  wedge, — and  that  the  joints, 
formed  by  the  contact  of  their  surfaces,  should  point  towards  a 
common  centre.  In  this  case,  no  one  portion  of  the  arch  can  be 
displaced  or  forced  inward ;  and  the  arch  cannot  be  broken  by 
any  force  which  is  not  sufficient  to  crush  the  materials  of  which 
it  is  made.  In  arches  made  of  common  bricks,  the  sides  of  which 
are  parallel,  any  one  of  the  bricks  might  be  forced  inward,  were 
it  not  for  the  adhesion  of  the  cement.  Any  two  of  the  bricks, 
however,  by  the  disposition  of  their  mortar,  cannot  collective¬ 
ly  be  forced  inward.  An  arch  of  the  proper  form,  when  com¬ 
plete,  is  rendered  stronger,  instead  of  weaker,  by  the  pressure  of 
a  considerable  weight,  provided  this  pressure  be  uniform.  While 
building,  however,  it  requires  to  be  supported  by  a  centring  of 
the  shape  of  its  internal  surface,  until  it  is  complete.  The  upper 
stone  of  an  arch  is  called  the  key-stone ,  but  is  not  more  essential 
than  any  other.  In  regard  to  the  shape  of  the  arch,  its  most 
simple  form  is  that  of  the  semi-circle.  It  is,  however,  very  fre¬ 
quently  a  smaller  arc  of  a  circle,  and,  still  more  frequently,  a  por¬ 
tion  of  an  ellipse.  The  simplest  theory  of  an  arch  supporting 
itself  only,  is  that  of  Dr.  Hooke.  The  arch,  when  it  has  only 
its  own  weight  to  bear,  may  be  considered  as  the  inversion  of  a 
chain,  suspended  at  each  end.  The  chain  hangs  in  such  a  form, 
that  the  weight  of  each  link  or  portion  is  held  in  equilibrium  by 


ARCHITECTURE. 


103 

the  result  of  two  forces  acting  at  its  extremities ;  and  these  forces, 
or  tensions,  are  produced,  the  one  by  the  weight  of  the  portion  of 
the  chain  below  the  link,  the  other  by  the  same  weight  increased 
by  that  of  the  link  itself,  both  of  them  acting  originally  in  a  ver¬ 
tical  direction.  Now,  supposing  the  chain  inverted,  so  as  to  con¬ 
stitute  an  arch  of  the  same  form  and  weight,  the  relative  situa¬ 
tions  of  the  forces  will  be  the  same,  only  they  will  act  in  contrary 
directions,  so  that  they  are  compounded  in  a  similar  manner,  and 
balance  each  other  on  the  same  conditions. 

The  arch  thus  formed  is  denominated  a  catenary  arch.  In 
common  cases,  it  differs  but  little  from  a  circular  arch  of  the  extent 
of  about  one  third  of  a  whole  circle,  and  rising  from  the  abut¬ 
ments  with  an  obliquity  of  about  30  degrees  from  a  perpendicu¬ 
lar.  But  though  the  catenary  arch  is  the  best  form  for  support¬ 
ing  its  own  weight,  and  also  all  additional  weight  which  presses 
in  a  vertical  direction,  it  is  not  the  best  form  to  resist  lateral 
pressure,  or  pressure  like  that  of  fluids,  acting  equally  in  all  direc¬ 
tions.  Thus  the  arches  of  bridges  and  similar  structures,  when 
covered  with  loose  stones  and  earth,  are  pressed  sideways,  as  well 
as  vertically,  in  the  same  manner  as  if  they  supported  a  weight 
of  fluid.  In  this  case,  it  is  necessary  that  the  arch  should  arise 
more  perpendicularly  from  the  abutment,  and  that  its  general 
figure  should  be  that  of  the  longitudinal  segment  of  an  ellipse. 
In  small  arches,  in  common  buildings,  where  the  disturbing 
force  is  not  great,  it  is  of  little  consequence  what  is  the  shape  of 
the  curve.  The  outlines  may  even  be  perfectly  straight,  as  in  the 
tier  of  bricks  which  we  frequently  see  over  a  window.  This  is, 
strictly  speaking,  a  real  arch,  provided  the  surfaces  of  the  bricks 
’tend  towards  a  common  centre.  It  is  the  weakest  kind  of  arch, 
and  a  part  of  it  is  necessarily  superfluous,  since  no  greater  portion 
can  act  in  supporting  a  weight  above  it,  than  can  be  included  be¬ 
tween  two  curved  or  arched  lines. 

Besides  the  arches  already  mentioned,  various  others  are  in  use. 
The  acute  or  lancet  arch,  much  used  in  Gothic  architecture,  is 


104 


AMERICAN  HOUSE-CARPENTER. 


described  usually  frorft  two  centres  outside  the  arch.  It  is  a 
strong  arch  for  supporting  vertical  pressure.  The  rampant  arch 
is  one  in  which  the  two  ends  spring  from  unequal  heights.  The 
horse-slioe  or  Moorish  arch  is  described  from  one  or  more  centres 
placed  above  the  base  line.  In  this  arch,  the  lower  parts  are  in 
danger  of  being  forced  inward.  The  ogee  arch  is  concavo-con¬ 
vex,  and  therefore  fit  only  for  ornament.  In  describing  arches, 
the  upper  surface  is  called  the  extrados ,  and  the  inner,  the  in- 
trados.  The  springing  lines  are  those  where  the  intrados  meets 
the  abutments,  or  supporting  walls.  The  span  is  the  distance 
from  one  springing  line  to  the  other.  The  wedge-shaped  stones, 
which  form  an  arch,  are  sometimes  called  voussoirs,  the  upper¬ 
most  being  the  key-stone.  The  part  of  a  pier  from  which  an 
arch  springs  is  called  the  impost ,  and  the  curve  formed  by  the 
upper  side  of  the  voussoirs,  the  archivolt.  It  is  necessary  that 
the  walls,  abutments  and  piers,  on  which  arches  are  supported, 
should  be  so  firm  as  to  resist  the  lateral  thrust ,  as  well  as  vertical 
pressure,  of  the  arch.  It  will  at  once  be  seen,  that  the  lateral  or 
sideway  pressure  of  an  arch  is  very  considerable,  when  we  recol¬ 
lect  that  every  stone,  or  portion  of  the  arch,  is  a  wedge,  a  part  of 
whose  force  acts  to  separate  the  abutments.  For  want  of  atten¬ 
tion  to  this  circumstance,  important  mistakes  have  been  committed, 
the  strength  of  buildings  materially  impaired,  and  their  ruin  ac¬ 
celerated.  In  some  cases,  the  want  of  lateral  firmness  in  the 
walls  is  compensated  by  a  bar  of  iron  stretched  across  the  span  of 
the  arch,  and  connecting  the  abutments,  like  the  tie-beam  of  a 
roof.  This  is  the  case  in  the  cathedral  of  Milan  and  some  other 
Gothic  buildings. 

In  an  arcade,  or  continuation  of  arches,  it  is  only  necessary  that 
the  outer  supports  of  the  terminal  arches  should  be  strong  enough 
to  resist  horizontal  pressure.  In  the  intermediate  arches,  the  lat¬ 
eral  force  of  each  arch  is  counteracted  by  the  opposing  lateral 
force  of  the  one  contiguous  to  it.  In  bridges,  however,  where 
individual  arches  are  liable  to  be  destroyed  by  accident,  it  is  desi- 


ARCHITECTURE. 


105 


table  that  each  of  the  piers  should  possess  sufficient  horizontal 
strength  to  resist  the  lateral  pressure  of  the  adjoining  arches. 

226.  — The  Yault  is  the  lateral  continuation  of  an  arch,  serving 
to  cover  an  area  or  passage,  and  bearing  the  same  relation  to  the 
arch  that  the  wall  does  to  the  column.  A  simple  vault  is  con¬ 
structed  on  the  principles  of  the  arch,  and  distributes  its  pressure 
equally  along  the  walls  or  abutments.  A  complex  or  groined 
vault  is  made  by  two  vaults  intersecting  each  other*  in  which 
case  the  pressure  is  thrown  upon  springing  points,  and  is  greatly 
increased  at  those  points.  The  groined  vault  is  common  in 
Gothic  architecture. 

227.  — The  Dome,  sometimes  called  cupola ,  is  a  concave  cover¬ 
ing  to  a  building,  or  part  of  it,  and  may  be  either  a  segment  of  a 
sphere,  of  a  spheroid,  or  of  any  similar  figure.  When  built  of 
stone,  it  is  a  very  strong  kind  of  structure,  even  more  so  than  the 
arch,  since  the  tendency  of  each  part  to  fall  is  counteracted,  not 
only  by  those  above  and  below  it,  but  also  by  those  on  each  side. 
It  is  only  necessary  that  the  constituent  pieces  should  have  a 
common  form,  and  that  this  form  should  be  somewhat  like  the 
frustum  of  a  pyramid,  so  that,  when  placed  in  its  situation,  its 
four  angles  may  point  toward  the  centre,  or  axis,  of  the  dome, 
During  the  erection  of  a  dome,  it  is  not  necessary  that  it  should 
be  supported  by  a  centring*  until  complete,  as  is  done  in  the  arch. 
Each  circle  of  stones,  when  laid,  is  capable  of  supporting  itself 
without  aid  from  those  above  it.  It  follows  that  the  dome  may 
be  left  open  at  top,  without  a  kev-stone,  and  yet  be  perfectly 
secure  in  this  respect,  being  the  reverse  of  the  arch.  The  dome 
of  the  Pantheon,  at  Rome,  has  been  always  open  at  top,  and  yet 
has  stood  unimpaired  for  nearly  2000  years.  The  upper  circle 
of  stones,  though  apparently  the  weakest,  is  nevertheless  often 
made  to  support  the  additional  weight  of  a  lantern  or  tower  above 
it.  In  several  of  the  largest  cathedrals,  there  are  two  domes,  one 
within  the  other,  which  contribute  their  joint  support  to  the  lan¬ 
tern,  which  rests  upon  the  top.  In  these  puddings,  the  dome 

14 


106 


AMERICAN  HOUSE-CARPENTER. 


rests  upon  a  circular  wall,  which  is  supported,  in  its  turn,  by 
arches  upon  massive  pillars  or  piers.  This  construction  is  called 
building  upon  pendentives ,  and  gives  open  space  and  loom  for 
passage  beneath  the  dome.  The  remarks  which  have  been  made 
in  regard  to  the  abutments  of  the  arch,  apply  equally  to  the  walls 
immediately  supporting  a  dome.  They  must  be  of  sufficient 
thickness  and  solidity  to  resist  the  lateral  pressure  of  the  dome, 
which  is  very  great.  The  walls  of  the  Roman  Pantheon  are  of 
great  depth  and  solidity.  In  order  that  a  dome  in  itself  should  be 
perfectly  secure,  its  lower  parts  must  not  be  too  nearly  vertical, 
since,  in  this  case,  they  partake  of  the  nature  of  perpendicular 
walls,  and  are  acted  upon  by  the  spreading  force  of  the  parts  above 
them.  The  dome  of  St.  Paul’s  church,  in  London,  and  some 
others  of  similar  construction,  are  bound  with  chains  or  hoops  of 
iron,  to  prevent  them  from  spreading  at  bottom.  Domes  which 
are  made  of  wood  depend,  in  part,  for  their  strength,  on  their  in¬ 
ternal  carpentry.  The  Halle  du  Bled,  in  Paris,  had  originally  a 
wooden  dome  more  than  200  feet  in  diameter,  and  only  one  foot 
in  thickness.  This  has  since  been  replaced  by  a  dome  of  iron. 
(See  Art.  303.) 

228. — The  Roof  is  the  most  common  and  cheap  method  of 
covering  buildings,  to  protect  them  from  rain  and  other  effects  of 
the  weather.  It  is  sometimes  flat,  but  more  frequently  oblique,  in 
its  shape.  The  flat  or  platform-roof  is  the  least  advantageous  for 
shedding  rain,  and  is  seldom  used  in  northern  countries.  The 
pent  roof,  consisting  of  two  oblique  sides  meeting  at  top,  is  the 
most  common  form.  These  roofs  are  made  steepest  in  cold  cli¬ 
mates,  where  they  are  liable  to  be  loaded  with  snow.  Where  the 
four  sides  of  the  roof  are  all  oblique,  it  is  denominated  a  hipped 
roof,  and  where  there  are  two  portions  to  the  roof,  of  different  ob¬ 
liquity,  it  is  a  curb,  or  mansard  roof.  In  modern  times,  roofs 
are  made  almost  exclusively  of  wood,  though  frequently  covered 
with  incombustible  materials.  The  internal  structure  or  carpen¬ 
try  of  roofs  is  a  subject  of  considerable  mechanical  contrivance. 


ARCHITECTURE. 


107 


The  roof  is  supported  by  rafters ,  which  abut  on  the  walls  on 
each  side,  like  the  extremities  of  an  arch.  If  no  other  timbers 
existed,  except  the  rafters,  they  would  exert  a  strong  lateral  pres¬ 
sure  on  the  walls,  tending  to  separate  and  overthrow  them.  To 
counteract  this  lateral  force,  a  tie-beam ,  as  it  is  called,  extends 
across,  receiving  the  ends  of  the  rafters,  and  protecting  the  wall 
from  their  horizontal  thrust.  To  prevent  the  tie-hpam  from 
sagging ,  or  bending  downward  with  its  own  weight,  a  king- 
jmst  is  erected  from  this  beam,  to  the  upper  angle  of  the  rafters, 
serving  to  connect  the  whole,  and  to  suspend  the  weight  of  the 
beam.  This  is  called  trussing.  Queeti-posts  are  sometimes 
added,  parallel  to  the  king-post,  in  large  roofs  ;  also  various  other 
connecting  timbers.  In  Gothic  buildings,  where  the  vaults  do 
not  admit  of  the  use  of  a  tie-beam,  the  rafters  are  prevented  from 
spreading,  as  in  an  arch,  by  the  strength  of  the  buttresses. 

In  comparing  the  lateral  pressure  of  a  high  roof  with  that  of  a 
low  one,  the  length  of  the  tie-beam  being  the  same,  it  will  be 
seen  that  a  high  roof,  from  its  containing  most  materials,  may 
produce  the  greatest  pressure,  as  far  as  weight  is  concerned.  On 
the  other  hand,  if  the  weight  of  both  be  equal,  then  the  low  roof 
will  exert  the  greater  pressure ;  and  this  will  increase  in  propor¬ 
tion  to  the  distance  of  the  point  at  which  perpendiculars,  drawn 
from  the  end  of  each  rafter,  would  meet.  In  roofs,  as  well  as  in 
wooden  domes  and  bridges,  the  materials  are  subjected  to  an  in¬ 
ternal  strain,  to  resist  which,  the  cohesive  strength  of  the  material 
is  relied  on.  On  this  account,  beams  should,  when  possible,  be 
of  one  piece.  Where  this  cannot  be  effected,  two  or  more  beams 
are  connected  together  by  splicing.  Spliced  beams  are  never  so 
strong  as  whole  ones,  yet  they  may  be  made  to  approach  the  same 
strength,  by  affixing  lateral  pieces,  or  by  making  the  ends  overlay 
each  other,  and  connecting  them  with  bolts  and  straps  of  iron. 
The  tendency  to  separate  is  also  resisted,  by  letting  the  two  pieces 
into  each  other  by  the  process  called  scarfing.  Mortices ,  in* 


108 


AMERICAN  HOUSE-CARPENTER. 


tended  to  truss  or  suspend  one  piece  by  another,  should  be  formed 
upon  similar  principles. 

Roofs  in  the  United  States,  after  being  boarded,  receive  a  se¬ 
condary  covering  of  shingles.  When  intended  to  be  incombustible, 
they  are  covered  with  slates  or  earthern  tiles,  or  with  sheets  of  lead, 
copper  or  tinned  iron.  Slates  are  preferable  to  tiles,  being  lighter, 
and  absorbing  less  moisture.  Metallic  sheets  are  chiefly  used  for 
flat  roofs,  wooden  domes,  and  curved  and  angular  surfaces,  which 
require  a  flexible  material  to  cover  them,  or  have  not  a  sufficient 
pitch  to  shed  the  rain  from  slates  or  shingles.  Various  artificial 
compositions  are  occasionally  used  to  cover  roofs,  the  most  com¬ 
mon  of  which  are  mixtures  of  tar  with  lime,  and  sometimes  with 
sand  and  gravel.” — Ency.  Ajn.  (See  Art.  285.) 


SECTION  III.— MOULDINGS,  CORNICES,  &c. 


MOULDINGS. 


229. — A  moulding  is  so  called,  because  of  its  being  of  the 
same  determinate  shape  along  its  whole  length,  as  though  the 
whole  of  it  had  been  cast  in  the  same  mould  or  form.  The  regular 
mouldings,  as  found  in  remains  of  ancient  architecture,  are  eight 
in  number  ;  and  are  known  by  the  following  names  : 


}  Annulet,  band,  cincture,  fillet,  listel  or  square. 


Fig.  123. 


Fig.  124. 


Astragal  or  bead. 


y  Torus  or  tore. 


Fig.  125. 


L 


Scotia,  trochilus  or  mouth. 


Fig.  126. 


Ovolo,  quarter-round  or  echinus. 


Fig.  127. 


110 


AMERICAN  HOUSE-CARPENTER. 


i'lfj.  138. 


Cavetto,  cove  or  hollow. 


Ogee. 


Some  of  the  terms  are  derived  thus :  fillet,  from  the  French 
word  fil,  thread.  Astragal,  from  astragalos ,  a  bone  of  the  heel 
— or  the  curvature  of  the  heel.  Bead,  because  this  moulding, 
when  properly  carved,  resembles  a  string  of  beads.  Torus,  or 
tore,  the  Greek  for  rope ,  which  it  resembles,  when  on  the  base  of 
a  column.  Scotia,  from  shotia,  darkness,  because  of  the  strong 
shadow  which  its  depth  produces,  and  which  is  increased  by  the 
projection  of  the  torus  above  it.  Ovolo,  from  ovum ,  an  egg, 
which  this  member  resembles,  when  carved,  as  in  the  Ionic  capi¬ 
tal.  Cavetto,  from  cavils ,  hollow.  Cymatium,  from  kumaton , 
a  wave. 

230.— Neither  of  these  mouldings  is  peculiar  to  any  one  of  the 
orders  of  architecture,  but  each  one  is  common  to  all ;  and  al¬ 
though  each  has  its  appropriate  use,  yet  it  is  by  no  means  con¬ 
fined  to  any  certain  position  in  an  assemblage  of  mouldings. 
The  use  of  the  fillet  is  to  bind  the  parts,  as  also  that  of  the  astra¬ 
gal  and  torus,  which  resemble  ropes.  The  ovolo  and  cyma-re- 
versa  are  strong  at  their  upper  extremities,  and  are  therefore  used 
to  support  projecting  parts  above  them.  The  cyma-recta  and 
cavetto,  being  weak  at  their  upper  extremities,  are  not  used  as 
supporters,  but  are  placed  uppermost  to  cover  and  shelter  the 
other  parts.  The  scotia  is  introduced  in  the  base  of  a  column,  to 


MOULDINGS,  CORNICES,  &C.  Ill 

separate  the  upper  and  lower  torus,  and  to  produce  a  pleasing 
variety  and  relief.  The  form  of  the  bead,  and  that  of  the  torus, 
is  the  same ;  the  reasons  for  giving  distinct  names  to  them  are, 
that  the  torus,  in  every  order,  is  always  considerably  larger  than 
the  bead,  and  is  placed  among  the  base  mouldings,  whereas  the 
bead  is  never  placed  there,  but  on  the  capital  or  entablature;  the 
torus,  also,  is  never  carved,  whereas  the  bead  is  ;  and  while  the 
torus  among  the  Greeks  is  frequently  elliptical  in  its  form,  the 
bead  retains  its  circular  shape.  While  the  scotia  is  the  reverse  of 
the  torus,  the  cavetto  is  the  reverse  of  the  ovolo,  and  the  cyma- 
recta  and  cyma-reversa  are  combinations  of  the  ovolo  and  cavetto. 

231.  — The  curves  of  mouldings,  in  Roman  architecture,  were 
most  generally  composed  of  parts  of  circles;  while  those  of  the 
Greeks  were  almost  always  elliptical,  or  of  some  one  of  the  conic 
sections,  but  rarely  circular,  except  in  the  case  of  the  bead,  which 
was  always,  among  both  Greeks  and  Romans,  of  the  form  of  a 
semi-circle.  Sections  of  the  cone  afford  a  greater  variety  of 
forms  than  those  of  the  sphere ;  and  perhaps  this  is  one  reason 
why  the  Grecian  architecture  so  much  excels  the  Roman.  The 
quick  turnings  of  the  ovolo  and  cyma-reversa,  in  particular,  when 
exposed  to  a  bright  sun,  cause  those  narrow,  well-defined  streaks 
of  light,  which  give  life  and  splendour  to  the  whole. 

232.  — A  profile  is  an  assemblage  of  essential  parts  and  mould¬ 
ings.  That  profile  produces  the  happiest  effect  which  is  com¬ 
posed  of  but  few  members,  varied  in  form  and  size,  and  arranged 
so  that  the  plane  and  the  curved  surfaces  succeed  each  other  al¬ 
ternately. 

233.  —  To  describe  the  Grecian  torus  and  scotia.  Join  the 
extremities,  a  and  6,  (j Fig.  131 ;)  and  from  fi  the  given  projection 
of  the  moulding,  draw/o,  at  right  angles  to  the  fillets;  from  b, 
draw  6  h,  at  right  angles  to  a  b  ;  bisect  a  b  in  c  ;  join  /  and  c, 
and  upon  c,  with  the  radius,  c  f,  describe  the  arc,  f  h,  cutting  b  h 
in  h  ;  through  c,  draw  d  e,  parallel  with  the  fillets  ;  make  d  c  and 
c  e,  each  equal  to  b  h  ;  then  d  e  and  a  b  will  be  conjugate  diame- 


112 


AMERICAN  HOUSE-CARPENTEfc* 


Fig.  131. 

ters  of  the  required  ellipse.  To  describe  the  curve  by  intersec¬ 
tion  of  lines,  proceed  as  directed  at  Art.  118  and  note  ;  by  a 
trammel,  see  Art.  125  ;  and  to  find  the  foci,  in  order  to  describe  it 
with  a  string,  see  Art.  115. 


d 

d 

~~ _ s/f 

/ 

c 

1 

c 

a 

b 

a 

Fig.  132.  Fig.  133 


234. — Fig.  132  to  139  exhibit  various  modifications  of  the1 
Grecian  ovolo,  sometimes  called  echinus.  Fig.  132  to  136  are 


113 


MOULDINGS,  CORNICES,  &C. 


elliptical,  a  b  and  b  c  being  given  tangents  to  the  curve ;  parallel 
to  which,  the  semi-conjugate  diameters,  a  d  and  d  c,  are  drawn  * 
In  Fig.  132  and  133,  the  lines,  a  d  and  d  c,  are  semi-axes,  the 
tangents,  a  b  and  b  c,  being  at  right  angles  to  each  other.  To 
draw  the  curve,  see  Art.  113.  In  Fig.  137,  the  curve  is  para¬ 
bolical,  and  is  drawn  according  to  Art.  127.  In  Fig.  138  and  139, 
the  curve  is  hyperbolical,  being  described  according  to  Art.  128. 
The  length  of  the  transverse  axis,  a  b ,  being  taken  at  pleasure, 
in  order  to  flatten  the  curve,  a  b  should  be  made  short  in  propor¬ 
tion  to  a  c. 


\ 


15 


114 


AMERICAN  HOUSE-CARPENTER. 


235. —  To  describe  the  Grecian  cavetto,  {Fig.  140  and  141,) 
having  the  height  and  projection  given,  see  Art.  118. 


236. —  To  describe  the  Grecian  cyma-recta.  When  the  pro¬ 
jection  is  more  than  the  height,  as  at  Fig.  142,  make  a  b  equal 
to  the  height,  and  divide  abed  into  4  equal  parallelograms ; 
then  proceed  as  directed  in  note  to  Art.  118.  When  the  projec¬ 
tion  is  less  than  the  height,  draw  d  a ,  {Fig.  143,)  at  right  angles 


to  a  b  ;  complete  the  rectangle,  a  b  ( 
rectangles,  and  proceed  according  to 


22$ 

Fig.  144. 

/ 

237. — To  describe  the  Grecian 


d  ;  divide  this  into  4  equal 
Art.  118. 


cyma-reversa.  When  the 


115 


MOULDINGS,  CORNICES,  &C 


projection  is  more  than  the  height,  as  at  Fig.  144,  proceed  as  di¬ 
rected  for  the  last  figure ;  the  curve  being  the  same  as  that,  the 
position  only  being  changed.  When  the  projection  is  less  than 
the  height,  draw  a  d ,  {Fig.  145,)  at  right  angles  to  the  fillet ; 
make  a  d  equal  to  the  projection  of  the  moulding  :  then  proceed 
as  directed  for  Fig.  142. 

238. — Roman  mouldings  are  composed  of  parts  of  circles,  and 
have,  therefore,  less  beauty  of  form  than  the  Grecian.  The  bead 
and  torus  are  of  the  form  of  the  semi-circle,  and  the  scotia,  also, 
in  some  instances ;  but  the  latter  is  often  composed  of  two  quad- 
rants,  having  different  radii,  as  at  Fig.  14G  and  147,  which  re¬ 
semble  the  elliptical  curve.  The  ovolo  and  cavetto  are  generally 
a  quadrant,  but  often  less.  When  they  are  less,  as  at  Fig.  150, 
the  centre  is  found  thus  :  join  the  extremities,  a  and  b ,  and  bisect 
a  b  in  c  ;  from  c,  and  at  right  angles  to  a  5,  draw  c  d,  cutting  a 
level  line  drawn  from  a  in  d  ;  then  d  will  be  the  centre.  This 
moulding  projects  less  than  its  height.  When  the  projection  is 
more  than  the  height,  as  at  Fig.  152,  extend  the  line  from  c  until 


Fig.  146. 


Fig.  147. 


116 


AMERICAN  HOUSE-CARPENTER. 


Fig.  152.  Fig.  153. 


Fig.  151. 


Fig.  155. 


Fig.  157. 


MOULDINGS,  CORNICES,  &C. 


117 


Fig.  158.  Fig.  159. 


it  cuts  a  perpendicular  drawn  from  a ,  as  at  d ;  and  that  will  be  the 
centre  of  the  curve.  In  a  similar  manner,  the  centres  are  found 
for  the  mouldings  at  Fig.  147,  151,  153,  156,  157,  158  and  159. 
The  centres  for  the  curves  at  Fig.  160  and  161,  are  found  thus : 
bisect  the  line,  a  b,  at  c  ;  upon  a,  c  and  6,  successively,  with  a  c 
or  c  b  for  radius,  describe  arcs  intersecting  at  d  and  d  ;  then  those 
intersections  will  be  the  centres. 

239. — Fig.  162  to  169  represent  mouldings  of  modern  inven¬ 
tion.  They  have  been  quite  extensively  and  successfully  used  in 
inside  finishing.  Fig.  162  is  appropriate  for  a  bed-moulding 
under  a  low,  projecting  shelf,  and  is  frequently  used  under  man¬ 
tle-shelves.  The  tangent,  i  h,  is  found  thus  :  bisect  the  line,  a  b, 
at  c,  and  b  c  at  d;  from  d,  draw  d  e,  at  right  angles  to  e  b  ;  from 
b,  draw  b  /,  parallel  to  e  d  ;  upon  b ,  with  b  d  for  radius,  describe 
the  arc,  d  f;  divide  this  arc  into  7  equal  parts,  and  set  one  of  the 
parts  from  s,  the  limit  of  the  projection,  to  o  ;  make  o  h  equal  to 
o  e  ;  from  h,  through  c,  draw  the  tangent,  li  i  ;  divide  b  h,  h  c,  ci 
and  i  a ,  each  into  a  like  number  of  equal  parts,  and  draw  the  in- 


118 


AMERICAN  HOUSE-CARPENTER. 


Fig.  163. 


Fig.  167. 


Fig.  168, 


Fig.  169 


tersecting  lines  as  directed  at  Art.  89.  If  a  bolder  form  is  desired, 
draw  the  tangent,  i  h,  nearer  horizontal,  and  describe  an  elliptic 
curve  as  shown  in  Fig.  131,  164,  175  and  176.  Fig.  163  is 
much  used  on  base,  or  skirting  of  rooms,  and  in  deep. panelling. 
The  curve  is  found  in  the  same  manner  as  that  of  Fig.  162.  In 
this  case,  however,  where  the  moulding  has  so  little  projection 


120 


AMERICAN  HOUSE-CARPENTER. 


in  comparison  with  its  height,  the  point,  e,  being  found  as  in  the 
last  figure,  h  s  may  be  made  equal  to  5  e,  instead  of  o  e  as  in  the 
last  figure.  Fig.  164  is  appropriate  for  a  crown  moulding  of  a 
cornice.  In  this  figure  the  height  and  projection  are  given ;  the 
direction  of  the  diameter,  a  b,  drawn  through  the  middle  of 
the  diagonal,  e  /  is  taken  at  pleasure ;  and  d  c  is  parallel  to  a 
e.  To  find  the  length  of  d  c,  draw  b  h,  at  right  angles  to  a  b  ; 
upon  o,  with  o  f  for  radius,  describe  the  arc,/  h,  cutting  b  h  in 
h  ;  then  make  o  c  and  o  d,  each  equal  to  b  li*  To  draw  the  curve, 
see  note  to  Art.  118.  Fig.  165  to  169  are  peculiarly  distinct  from 
ancient  mouldings,  being  composed  principally  of  straight  lines  ; 
the  few  curves  they  possess  are  quite  short  and  quick. 


H.  P. 


5 

15 

4 

7 

121 

r\  y 

2 

III 

J 

9 

s  1 

10 

Fig.  170. 


H 

15 

31 

141 

7 

31 

/ 

— 

13 

qJ 

H 

li4 

7 

l 

1 

ih 

□ 

9 

10i 

10 

Fig.  171. 


240.— Fig.  170  and  171  are  designs  for  antse  caps.  The 

* 

*  The  manner  of  ascertaining  the  length  of  the  conjugate  diameter,  d  c,  in  this  figure, 
and  also  in  Fig.  131,  175  and  175,  is  new,  and  is  important  in  this  application.  It  is 
founded  upon  well-known  mathematical  principles,  viz :  All  the  parallelograms  that  may 
be  circumscribed  about  an  ellipsis  are  equal  to  one  another,  and  consequently  any  one 
is  equal  to  the  rectangle  of  the  two  axes.  And  again  :  the  sum  of  the  squares  of  every 
pair  of  conjugate  diameters  is  equal  to  the  sum  of  the  squares  of  the  two  axes. 


Mouldings,  cornices,  &c. 


I2i 

diameter  of  the  antae  is  divided  into  20  equal  parts,  and  the  height 
and  projection  of  the  members,  are  regulated  in  accordance  with 
those  parts,  as  denoted  under  H  and  P,  height  and  projection. 
The  projection  is  measured  from  the  middle  of  the  antae.  These 
will  be  found  appropriate  for  porticos,  door-ways,  mantle-pieces, 
door  and  window  trimmings,  &c.  The  height  of  the  antae  for 
mantle-pieces,  should  be  from  5  to  6  diameters,  having  an  entab¬ 
lature  of  from  2  to  2|  diameters.  This  is  a  good  proportion,  it 
being  similar  to  the  Doric  order.  But  for  a  portico  these  propor¬ 
tions  are  much  too  heavy  ;  an  antae,  15  diameters  high,  and  an  en¬ 
tablature  of  3  diameters,  will  have  a  better  appearance. 

CORNICES. 

241. — Fig.  172,  173  and  174,  are  designs  for  eave  cornices, 
and  Fig.  175  and  176,  for  stucco  cornices  for  the  inside  finish  of 
rooms.  The  projection  of  the  uppermost  member  from  the  facia, 
is  divided  into  20  equal  parts,  and  the  various  members  are  pro¬ 
portioned  according  to  those  parts,  as  figured  under  i7and  P. 


H.  P. 


1* 

20 

- - , 

1 

5 

U 

17J 

li 

J 

8 

16} 

. J  *  - 

1 

1 

J 

25 

Fif.  m. 
16 


122 


AMERICAN  HOUSE-CARPENTER 


H.  P. 


Fig.  174. 


123 


MOULDINGS,  CORNICES,  &C. 


Fig.  176, 


124 


AMERICAN  HOUSE-CARPENTER. 


242. —  To  proportion  an  eave  cornice  in  accordance  with  the 
height  of  the  building.  Draw  the  line,  a  c,  {Fig.  177,)  and 
make  b  c  and  b  a,  each  equal  to  18  inches ;  from  b ,  draw  b  d,  at 
right  angles  to  a  c,  and  equal  in  length  to  f  of  a  c;  bisect  b  d  in 
e,  and  from  a,  through  e,  draw  a  f;  upon  a,  with  a  c  for  radius, 
describe  the  arc,  c  f  and  upon  e,  with  e  f  for  radius,  describe  the 
arc,/ d  ;  divide  the  curve,  df  c,  into  7  equal  parts,  as  at  10,  20, 
30,  &c.,  and  from  these  points  of  division,  draw  lines  to  b  c,  pa¬ 
rallel  to  d  b  ;  then  the  distance,  b  1,  is  the  projection  of  a  cornice 
for  a  building  10  feet  high  ;  b  2,  the  projection  at  20  feet  high  ; 
b  3,  the  projection  at  30  feet,  &c.  If  the  projection  of  a  cornice  for 
a  building  34  feet  high,  is  required,  divide  the  arc  between  30  and 
40  into  10  equal  parts,  and  from  the  fourth  point  from  30,  draw  a 
line  to  the  base,  b  c,  parallel  with  b  d  ;  then  the  distance  of  the 
point,  at  which  that  line  cuts  the  base,  from  b,  will  be  the  projec¬ 
tion  required.  So  proceed  for  a  cornice  of  any  height  within  70 
feet.  The  above  is  based  on  the  supposition  that  18  inches  is  the 
proper  projection  for  a  cornice  70  feet  high.  This,  for  general 
purposes,  will  be  found  correct ;  still,  the  length  of  the  line,  b  c, 
may  be  varied  to  suit  the  judgment  of  those  who  think  differ¬ 
ently. 

Having  obtained  the  projection  of  a  cornice,  divide  it  into  20 
equal  parts,  and  apportion  the  several  members  according  to  its 
destination — as  is  shown  at  Fig.  172,  173  and  174, 


MOULDINGS,  CORNICES,  &C.  *  125 


Fig.  178. 

243.  —  To  'proportion  a  cornice  according  to  a  smaller  given 
one.  Let  the  cornice  at  Fig.  178  be  the  given  one.  Upon  any 
point  in  the  lowest  line  of  the  lowest  member,  as  at  a,  with  the 
height  of  the  required  cornice  for  radius,  describe  an  intersecting 
arc  across  the  uppermost  line,  as  at  b  ;  join  a  and  b  ;  then  b  1  will 
be  the  perpendicular  height  of  the  upper  fillet  for  the  proposed  cor¬ 
nice,  1  2  the  height  of  the  crown  moulding — and  so  of  all  the 
members  requiring  to  be  enlarged  to  the  sizes  indicated  on  this 
line.  For  the  projection  of  the  proposed  cornice,  draw  a  d ,  at  right 
angles  to  a  b,  and  c  d,  at  right  angles  to  be;  parallel  with  c  d, 
draw  lines  from  each  projection  of  the  given  cornice  to  the  line, 
ad;  then  e  d  will  be  the  required  projection  for  the  proposed 
cornice,  and  the  perpendicular  lines  falling  upon  e  d  will  indicate 
the  proper  projection  for  the  members. 

244.  —  To  proportion  a  cornice  according  to  a  larger  given 
one.  Let  A,  (Fig.  179,)  be  the  given  cornice.  Extend  a  o  to  6, 
and  draw  c  d ,  at  right  angles  to  a  b ;  extend  the  horizontal  lines 
of  the  cornice,  A,  until  they  touch  o  d;  place  the  height  of  the 
proposed  cornice  from  o  to  e,  and  join  /  and  e  ;  upon  o,  with  the 
projection  of  the  given  cornice,  o  a,  for  radius,  describe  the  quad¬ 
rant,  ad  ;  from  c?,  draw  d  b ,  parallel  to/e;  upon  o,  with  o  b  for 
radius,  describe  the  quadrant,  be;  then  o  c  will  be  the  proper  pro¬ 
jection  for  the  proposed  cornice.  Join  a  and  c ;  draw  lines  from  the 


126 


AMERICAN  HOUSE-CARPENTER. 


Fig.  179.  d 


projection  of  the  different  members  of  the  given  cornice  to  a  o, 
parallel  to  o  d;  from  these  divisions  on  the  line,  a  o,  draw  lines 
to  the  line,  o  c,  parallel  to  a  c  ;  from  the  divisions  on  the  line,  of, 
draw  lines  to  the  line,  o  e ,  parallel  to  the  line,  f  e  ;  then  the  di¬ 
visions  on  the  lines,  o  e  and  o  c,  will  indicate  the  proper  height  and 
projection  for  the  different  members  of  the  proposed  cornice.  In 
this  process,  we  nave  assumed  the  height,  o  e,  of  the  proposed 
cornice  to  be  given ;  but  if  the  projection,  o  c,  alone  be  given,  we 
can  obtain  the  same  result  by  a  different  process.  Thus:  upon  o, 
with  o  c  for  radius,  describe  the  quadrant,  cb  ;  upon  o,  with  o  a 
for  radius,  describe  the  quadrant,  ad  ;  join  d  and  b  ;  from  f  draw 
f  e,  parallel  to  d  b  ;  then  o  e  will  be  the  proper  height  for  the  pro¬ 
posed  cornice,  and  the  height  and  projection  of  the  different  mem¬ 
bers  can  be  obtained  by  the  above  directions.  By  this  problem, 
a  cornice  can  be  proportioned  according  to  a  smaller  given  one 
as  well  as  to  a  larger  ;  but  the  method  described  in  the  previous 
article  is  much  more  simple  for  that  purpose. 

245. —  To  find  the  angle-bracket  for  a  cornice.  Let  A,  (Fig. 
180,)  be  the  wall  of  the  building,  and  B  the  given  bracket,  which, 
for  the  present  purpose,  is  turned  down  horizontally.  The  angle- 
bracket,  C,  is  obtained  thus :  through  the  extremity,  a,  and  paral- 


MOULDINGS,  CORNICES,  &C. 


127 


lei  with  the  wall,/rf,  draw  the  line,  a  b  ;  make  e  c  equal  a  /, 
and  through  c,  draw  c  b,  parallel  with  e  d  ;  join  d  and  b,  and  from 
the  several  angular  points  in  B,  draw  ordinates  to  cut  d  b  in  1,  2 
and  3 ;  at  those  points  erect  lines  perpendicular  to  d  b  ;  from  h, 
draw  li  g ,  parallel  to  fa;  take  the  ordinates,  1  o,  2  o,  &c.,  at  B, 
and  transfer  them  to  C,  and  the  angle-bracket,  C,  will  be  defined. 
In  the  same  manner,  the  angle-bracket  for  an  internal  cornice,  or 
the  angle-rib  of  a  coved  ceiling,  or  of  groins,  as  at  Fig.  181,  can 
be  found. 

246. — A  level  crown  moulding  being  given,  to  find  the  raking 
moulding  and  a  level  return  at  the  top.  Let  A,  {Fig.  182,)  be 
the  given  moulding,  and  A  b  the  rake  of  the  roof.  Divide  the 
curve  of  the  given  moulding  into  any  number  of  parts,  equal  or 
unequal,  as  at  1,  2,  and  3  ;  from  these  points,  draw  horizontal 
lines  to  a  perpendicular  erected  from  c ;  at  any  convenient  place 
on  the  rake,  as  at  B ,  draw  a  c,  at  right  angles  to  A  b  ;  also,  from 
b,  draw  the  horizontal  line,  b  a  ;  place  the  thickness,  d  a ,  of  the 
moulding  at  A ,  from  b  to  a,  and  from  a,  draw  the  perpendicular 
line,  a  e  ;  from  the  points,  1,  2,  3,  at  A,  draw  lines  to  C,  parallel 
to  A  b ;  make  a  1,  a  2  and  a  3,  at  B  and  at  C,  equal  to  a  1,  &c., 
at  A  ;  through  the  points,  1,  2  and  3,  at  B,  trace  the  curve — this 
will  be  the  proper  form  for  the  raking  moulding.  From  1,  2  and 


128 


AMERICAN  HOUSE-CARPENTER. 


C 

a  321  t 


3,  at  C,  drop  perpendiculars  to  the  corresponding  ordinates  from 
1,  2  and  3,  at  A ;  through  the  points  of  intersection,  trace  the 
curve — this  will  be  the  proper  form  for  the  return  at  the  top. 


rf 


SECTION  IV.— FRAMING. 


247.— This  subject  is,  to  the  carpenter,  of  the  highest  impor¬ 
tance  ;  and  deserves  more  attention  and  a  larger  place  in  a  volume 
of  this  kind,  than  is  generally  allotted  to  it.  Something,  indeed,- 
has  been  said  upon  the  geometrical  principles,  by  which  the  seve¬ 
ral  lines  for  the  joints  and  the  lengths  of  timber,  may  be  ascer¬ 
tained  ;  yet,  besides  this,  there  is  much  to  be  learned.  For  how¬ 
ever  precise  or  workmanlike  the  joints  may  be  made,  what  will 
it  avail,  should  the  system  of  framing,  from  an  erroneous  position 
of  its  timbers,  &c.,  change  its  form,  or  become  incapable  of  sus¬ 
taining  eVen  its  own  weight?  Hence  the  necessity  for  a  know¬ 
ledge  of  the  laws  of  pressure  and  the  strength  of  timber.  These 
being  once  understood,  we  can  with  confidence  determine  the  best 
position  and  dimensions  for  the  several  timbers  which  compose  a 
floor  or  a  roof,  a  partition  or  a  bridge.  As  systems  of  framing 
are  more  or  less  exposed  to  heavy  weights  and  strains,  and,  in 
case  of  failure,  cause  not  only  a  loss  of  labour  and  material,  but 
frequently  that  of  life  itself,  it  is  very  important  that  the  materials 
employed  be  of  the  proper  quantity  and  quality  to  serve  their  des¬ 
tination.  And,  on  the  other  hand,  any  superfluous  material  is  not 
only  useless,  but  a  positive  injury,  it  being  an  unnecessary  load 
upon  the  points  of  support.  It  is  necessary,  therefore,  to  know 


17 


130 


AMERICAN  HOUSE-CARPENTER. 


the  least  quantity  of  limber  that  will  suffice  for  strength.  The 
greatest  fault  in  framing  is  that  of  using  an  excess  of  material. 
Economy,  at  least,  would  seem  to  require  that  this  evil  be  abated. 

Before  proceeding  to  consider  the  principles  upon  which  a  sys¬ 
tem  of  framing  should  be  constructed,  let  us  attend  to  a  few  of 
the  elementary  laws  in  Mechanics ,  which  will  be  found  to  be  of 
great  value  in  determining  those  principles. 

248. — Laws  of  Pressure.  (1.)  A  heavy  body  always 
exerts  a  pressure  equal  to  its  own  weight  in  a  vertical  direction. 
Example:  Suppose  an  iron  ball,  weighing  100  lbs.,  be  supported 
upon  the  top  of  a  perpendicular  post,  {Fig.  196;)  then  the 
pressure  exerted  upon  that  post  will  be  equal  to  the  weight  of  the 
ball ;  viz.,  100  lbs.  (2.)  But  if  two  inclined  posts,  {Fig.  183,) 
be  substituted  for  the  perpendicular  support,  the  united  pressures 
upon  these  posts  will  be  more  than  equal  to  the  weight,  and  will 
be  in  proportion  to  their  position.  The  farther  apart  their  feet  are 
spread  the  greater  will  be  the  pressure,  and  vice  versa.  Hence 
tremendous  strains  may  be  exerted  by  a  comparatively  small 
weight.  And  it  follows,  therefore,  that  a  piece  of  timber  intend¬ 
ed  for  a  strut  or  post,  should  be  so  placed  that  its  axis  may  coin¬ 
cide,  as’  near  as  possible,  with  the  direction  of  the  pressure.  The 
direction  of  the  pressure  of  the  weight,  W,  {Fig.  183,)  is  in  the 
vertical  line,  b  d  ;  and  the  weight,  W,  would  fall  in  that  line,  if 
the  two  posts  were  removed,  hence  the  best  position  for  a  support 


w 


FRAMING. 


131 


for  the  weight  would  be  in  that  line.  But,  as  it  rarely  occurs  in 
systems  of  framing  that  weights  can  be  supported  by  any  single 
resistance,  they  requiring  generally  two  or  more  supports,  (as  in 
the  case  of  a  roof  supported  by  its  rafters,)  it  becomes  important, 
therefore,  to  know  the  exact  amount  of  pressure  any  certain 
weight  is  capable  of  exerting  upon  oblique  supports.  This  can 
be  ascertained  by  the  following  process. 

Let  a  6  and  b  c,  (Fig.  183,)  represent  the  axes  of  two  sticks  of 
timber  supporting  the  weight,  TV;  and  let  the  weight,  W:  be 
equal  to  6  tons.  Make  the  vertical  line,  b  d,  equal  to  6  inches  ; 
from  d,  draw  df  parallel  to  a  b,  and  d  e,  parallel  to  cb  ;  then 
the  line,  b  e,  will  be  found  to  be  3h  inches  long,  which  is  equal  to 
the  number  of  tons  that  the  weight,  TV,  exerts  upon  the  post,  a  b . 
The  pressure  upon  the  other  post  is  represented  by  bf,  which  in 
this  case  is  of  the  same  length  as  b  e.  The  posts  being  inclined 
at  equal  angles  to  the  vertical  line,  b  d,  the  pressure  upon  them  is 
equal.  Thus  it  will  be  found  that  the  weight,  which  weighs 
only  6  tons,  exerts  a  pressure  of  7  tons  ;  the  amount  being  In¬ 
creased  because  of  the  oblique  position  of  the  supports.  The 
lines,  eb,bf,fd  and  d  e,  compose  what  is  called  the  'parallelo¬ 
gram  of  forces.  The  oblique  strains  exerted  by  any  one  force, 
therefore,  may  always  be  ascertained,  by  making  b  d  equal,  (upon 
any  scale  of  equal  parts,)  to  the  number  of  lbs.,  cwts.,  or  tons 
contained  in  the  weight,  W,  and  b  e  will  then  represent  the  num¬ 
ber  of  lbs.,  cwts.,  or  tons  with  which  the  timber,  a  b,  is  pressed, 
and  bf  that  exerted  upon  b  c. 


Fig.  184 


132 


AMERICAN  HOUSE-CARPENTER. 


Correct  ideas  of  the  comparative  pressure  exerted  upon  timbers 
according  to  their  position,  will  be  readily  formed  by  drawing 
various  designs  of  framing,  and  estimating  the  several  strains  in 
accordance  with  these  principles.  In  Fig.  184,  the  struts  are 
framed  into  a  third  piece,  and  the  weight  suspended  from  that. 
The  struts  are  placed  at  a  different  angle  to  show  the  diverse 
pressures.  The  length  of  the  timber  used  as  struts,  does  not 
alter  the  amount  of  the  pressure.  But  it  may  be  observed  that 
long  timbers  are  not  so  capable  of  resistance  as  short  ones. 


w 


249. — In  Fig.  185,  the  weight,  W,  exerts  a  pressure  on  the 
struts  in  the  direction  of  their  length  ;  their  feet,  n,  n,  have,  there¬ 
fore,  a  tendency  to  move  in  the  direction,  n  o,  and  would  so  move, 
were  they  not  opposed  by  a  sufficient  resistance  from  the  blocks, 
A  and  A.  If  a  piece  of  each  block  be  cut  off  at  the  horizontal 
line,  a  n,  the  feet  of  the  struts  would  slide  away  from  each  other 
along  that  line,  in  the  direction,  n  a  ;  but  if,  instead  of  these,  two 
pieces  were  cut  off  at  the  vertical  line,  n  h ,  then  the  struts  would 
descend  vertically.  To  estimate  the  horizontal  and  the  vertical 
pressures  exerted  by  the  struts,  let  n  o  be  made  equal  (upon  any 
scale  of  equal  parts)  to  the  number  of  tons  (or  pounds)  with 
which  the  strut  is  pressed  ;  construct  the  parallelogram  of  forces 


FRAMING. 


133 


by  drawing  o  e  parallel  to  a  n,  and  o  f  parallel  to  b  n  ;  then  n  /, 
(by  the  same  scale,)  shows  the  number  of  tons  (or  pounds)  pres¬ 
sure  that  is  exerted  by  the  strut  in  the  direction,  n  a ,  and  n  e 
shows  the  amount  exerted  in  the  direction,  n  b.  By  constructing 
designs  similar  to  this,  giving  various  and  dissimilar  positions  to 
the  struts,  and  then  estimating  the  pressures,  it  will  be  found  in 
every  case  that  the  horizontal  pressure  of  one  strut  is  exactly 
equal  to  that  of  the  other,  however  much  one  strut  may  be  in¬ 
clined  more  than  the  other ;  and  also,  that  the  united  vertical 
pressure  of  the  two  struts  is  exactly  equal  to  the  weight,  W.  (In 
this  calculation,  the  weight  of  the  timbers  is  not  taken  into  con¬ 
sideration.) 

250. — Suppose  that  the  two  struts,  B  and  B,  (Fig.  185,)  were 
rafters  of  a  roof,  and  that  instead  of  the  blocks,  A  and  A,  the  walls 
of  a  building  were  the  supports  :  then,  to  prevent  the  walls  from 
being  thrown  over  by  the  thrust  of  B  and  B,  it  would  be  desira¬ 
ble  to  remove  the  horizontal  pressure.  This  may  be  done  by  uni¬ 
ting  the  feet  of  the  rafters  with  a  rope,  iron  rod,  or  piece  of  tim¬ 
ber,  as  in  Fig.  186.  This  figure  is  similar  to  the  truss  of  a  roof. 


The  horizontal  strains  on  the  tie-beam,  tending  to  pull  it  asunder 
in  the  direction  of  its  length,  may  be  measured  at  the  foot  of  the 


134 


AMERICAN  HOUSE-CARPENTER. 


rafter,  as  was  shown  at  Fig.  185 ;  but  it  can  be  more  readily 
and  as  accurately  measured,  by  drawing  from /and  e  horizontal 
lines  to  the  vertical  line,  b  d ,  meeting  it  in  o  and  o  ;  then/  o  will  be 
the  horizontal  thrust  at  B,  and  e  o  at  A  ;  these  will  be  found  to 
equal  one  another.  When  the  rafters  of  a  roof  are  thus  connected, 
all  tendency  to  thrust  the  walls  horizontally  is  removed,  the  only 
pressure  on  them  is  in  a  vertical  direction,  being  equal  to  the 
weight  of  the  roof  and  whatever  it  has  to  support.  This  pres¬ 
sure  is  beneficial  rather  than  otherwise,  as  a  roof  thus  formed 
tends  to  steady  the  walls. 


251. — Fig.  187  and  188  exhibit  methods  of  framing  for  sup¬ 
porting  the  equal  weights,  TV  and  TV.  Suppose  it  be  required  to 
measure  and  compare  the  strains  produced  on  the  pieces,  A  B 
and  A  C.  Construct  the  parallelogram  of  forces,  e  b  f  d,  ac¬ 
cording  to  Art.  248.  Then  b  f  show  will  the  strain  on  A  B ,  and  b 
e  the  strain  on  A  C.  By  comparing  the  figures,  b  d  being  equal 
in  each,  it  will  be  seen  that  the  strains  in  Fig.  187  are  about  three 


FRAMING. 


135 


times  as  great  as  those  in  Fig.  188 :  the  position  of  the  pieces, 
A  B  and  A  C:  in  Fig.  188,  is  therefore  far  preferable. 

This  and  the  preceding  examples  exemplify,  in  a  measure,  the 
resolution  of  forces  ;  viz.,  the  finding  of  two  or  more  forces,  which, 
acting  in  different  directions,  shall  exactly  balance  the  pressure 
of  any  given  single  force.  Thus,  in  Fig.  185,  supposing  the 
weight,  W,  to  be  the  greatest  force  that  the  two  timbers,  in  their 
present  position,  are  capable  of  sustaining,  then  the  weight,  W, 
is  the  given  force,  and  the  timbers  are  the  two  forces  just  equal  to 
the  given  force. 


252. — The  composition  of  forces  consists  in  ascertaining  the 
direction  and  amount  of  one  force,  which  shall  be  just  capable  of 
balancing  two  or  more  given  forces,  acting  in  different  directions. 
This  is  only  the  reverse  of  the  resolution  of  forces,  and  the  two 
are  founded  on  one  and  the  same  principle,  and  may  be  solved  in 
the  same  maimer.  For  example;  let  A  and  B,  {Fig.  189,)  be 
two  pieces  of  timber,  pressed  in  the  direction  of  their  length  to¬ 
wards  b — A  by  a  force  equal  to  6  tons  weight,  and  B  equal  to  9. 
To  find  the  direction  and  amount  of  pressure  they  would  unitedly 
exert,  draw  the  lines,  b  e  and  b  f  in  a  line  with  the  axes  of  the 
timbers,  and  make  b  e  equal  to  the  pressure  exerted  by  B.  viz.,  9  ; 
also  make  b  f  equal  to  the  pressure  on  A,  viz.,  6,  and  complete 
the  parallelogram  of  forces,  eb  f  d;  then  b  d,  the  diagonal  of  the 


136 


AMERICAN  HOUSE-CARPENTER. 


parallelogram,  will  be  the  direction ,  and  its  length  will  be  the 
amount ,  of  the  united  pressures  of  A  and  of  B.  The  line,  b  d,  is 
termed  the  resultant  of  the  two  forces,  b  f  and  be.  If  A  and  B  are  to 
be  supported  by  one  post,  C ,  the  best  position  for  that  post  will  be 
in  the  direction  of  the  diagonal,  b  d;  and  it  will  require  to  be 
sufficiently  strong  to  support  the  united  pressures  of  A  and  of  B. 


253. — Another  example:  let  Fig.  190  represent  a  piece  of 
framing  commonly  called  a  crane,  which  is  used  for  hoisting 
heavy  weights  by  means  of  the  rope,  B  b  f,  which  passes  over  a 
pulley  at  b.  This  is  similar  to  Fig.  187  and  188,  yet  it  is  mate¬ 
rially  different.  In  those  figures,  the  strain  is  in  one  direction  • 
only,  vis.,  from  b  to  d  ;  but  in  this  there  are  two  strains,  from  A 
to  B  and  from  A  to  W.  The  strain  in  the  direction,  A  B,  is  evi¬ 
dently  equal  to  that  in  the  direction,  A  W.  To  ascertain  the  best 
position  for  the  strut,  A  C ,  make  b  e  equal  to  b  /,  and  complete 
the  parallelogram  of  forces,  e  bf  d  ;  then  draw  the  diagonal,  b  d, 
and  it  will  be  the  position  required.  Should  the  foot,  C,  of  the 
strut  be  placed  either  higher  or  lower,  the  strain  on  A  C  would  be 
increased.  In  constructing  cranes,  it  is  advisable,  in  order  that 
the  piece,  B  A,  may  be  under  a  gentle  pressure,  to  place  the  foot 
of  the  strut  a  trifle  lower  than  where  the  diagonal,  6  d,  would  in¬ 
dicate,  but  never  higher 


FRAMING. 


137 


Fig.  191.  W 

254. —  Ties  and  Struts.  Timbers  in  a  state  of  tension  are 
called  ties,  while  such  as  are  in  a  state  of  compression  are  termed 
struts.  This  subject  can  be  illustrated  in  the  following  manner. 

Let  A  and  B ,  {Fig.  191,)  represent  beams  of  timber  supporting 
the  weights,  W,  W  and  W ;  A  having  but  one  support,  which  is 
in  the  middle  of  its  length,  and  B  two,  one  at  each  end.  To 
show  the  nature  of  the  strains,  let  each  beam  be  sawed  in  the 
middle  from  a  to  b.  The  effects  are  obvious :  the  cut  in  the 
beam,  A,  will  open,  whereas  that  in  B  will  close.  If  the  weights 
are  heavy  enough,  the  beam,  A,  will  break  at  b  ;  while  the  cut  in 
B  will  be  closed  perfectly  tight  at  a,  and  the  beam  be  very  little 
injured  by  it.  But  if,  on  the  other  hand,  the  cuts  be  made  in  the 
bottom  edge  of  the  timbers,  from  c  to  b,  B  will  be  seriously  in¬ 
jured,  while  A  will  scarcely  be  affected.  By  this  it  appears  evident 
that,  in  a  piece  of  timber  subject  to  a  pressure  across  the  direction 
of  its  length,  the  fibres  are  exposed  to  contrary  strains.  If  the  tim¬ 
ber  is  supported  at  both  ends,  as  at  B,  those  from  the  top  edge  down 
to  the  middle  are  compressed  in  the  direction  of  their  length,  while 
those  from  the  middle  to  the  bottom  edge  are  in  a  state  of  tension  ; 
but  if  the  beam  is  supported  as  at  A,  the  contrary  effect  is  produced ; 
while  the  fibres  at  the  middle  of  either  beam  are  not  at  all  strained. 
The  strains  in  a  framed  truss  are  of  the  same  nature  as  those  in 
a  single  beam.  The  truss  for  a  roof,  being  supported  at  each  end, 
has  its  tie-beam  in  a  state  of  tension,  while  its  rafters  are  com¬ 
pressed  in  the  direction  of  their  length.  By  this,  it  appears  highly 
important  that  pieces  in  a  state  of  tension  should  be  distinguished 

18 


138 


AMERICAN  HOUSE-CARPENTER. 


from  such  as  are  compressed,  in  order  that  the  former  may  be  pre¬ 
served  continuous.  A  strut  may  be  constructed  of  two  or  more 
pieces;  yet,  where  there  are  many  joints,  it  will  not  resist  com¬ 
pression  so  firmly. 

255.  —  To  distinguish  ties  from  struts.  This  may  be  done 
by  the  following  rule.  In  Fig.  183,  the  timbers,  a  b  and  b  c,  are  the 
sustaining  forces,  and  the  weight,  TV,  is  the  straining  force  ;  and, 
if  the  support  be  removed,  the  straining  force  would  move  from 
the  point  of  support,  5,  towards  d.  Let  it  be  required  to  ascer- 

t 

tain  whether  the  sustaining  forces  are  stretched  or  pressed  by  the 
straining  force.  Rule  :  upon  the  direction  of  the  straining  force, 
b  d,  as  a  diagonal,  construct  a  parallelogram,  e  b  f  d,  whose  sides 
shall  be  parallel  with  the  direction  of  the  sustaining  forces,  a  b 
andci;  through  the  point,  b,  draw  a  line,  parallel  to  the  diag¬ 
onal,  e  f ;  this  may  then  be  called  the  dividing  line  between  ties 
and  struts.  Because  all  those  supports  which  are  on  that  side  of 
the  dividing  line,  which  the  straining  force  would  occupy  if  unre¬ 
sisted,  are  compressed,  while  those  on  the  other  side  of  the  divi¬ 
ding  line  are  stretched. 

In  Fig.  183,  the  supports  are  both  compressed,  being  on  that 
side  of  the  dividing  line  which  the  straining  force  would  occupy 
if  unresisted.  In  Fig.  187  and  188,  in  which  A  B  and  A  C 
are  the  sustaining  forces,  A  C  is  compressed,  whereas  A  B  is  in 
a  state  of  tension  ;  A  C  being  on  that  side  of  the  line,  h  i ,  which 
the  straining  force  would  occupy  if  unresisted,  and  A  B  on  the 
opposite  side.  The  place  of  the  latter  might  be  supplied  by  a 
chain  or  rope.  In  Fig.  186,  the  foot  of  the  rafter  at  A  is  sus¬ 
tained  by  two  forces,  the  wall  and  the  tie-beam,  one  perpendicular 
and  the  other  horizontal :  the  direction  of  the  straining  force  is 
indicated  by  the  line,  b  a.  The  dividing  line,  h  i,  ascertained 
by  the  rule,  shows  that  the  wall  is  pressed  and  the  tie-beam 
stretched. 

256.  — Another  example  :  let  E  A  B  F,  [Fig.  192,)  represent 
a  gate,  supported  by  hinges  at  A  and  E.  In  this  case,  the  strait *■ 


FRAMING. 


139 


Fig.  192. 


ing  force  is  the  weight  of  the  materials,  and  the  direction  of 
course  vertical.  Ascertain  the  dividing  line  at  the  several  points, 
G,  B ,  /,  f  H  and  F.  It  will  then  appear  that  the  force  at  G  is 
sustained  by  A  G  and  G  E ,  and  the  dividing  line  shows  that  the 
former  is  stretched  and  the  latter  compressed.  The  force  at  H  is 
supported  by  A  H and  HE — the  former  stretched  and  the  latter 
compressed.  The  force  at  B  is  opposed  by  H  B  and  A  B ,  one 
pressed — the  other  stretched.  The  force  at /Ts  sustained  by  G 
i^and  F  E,  G  F  being  stretched  and  F  E  pressed.  By  this  it 
appears  that  A  B  is  in  a  state  of  tension,  and  E  F,  of  compres¬ 
sion;  also,  that  A  H  and  G  F  are  stretched,  while  B  H  and  G 
E  are  compressed :  which  shows  the  necessity  of  having  A  H 
and  G  F \  each  in  one  whole  length,  while  B  H and  G  E  may 
be,  as  they  are  shown,  each  in  two  pieces.  The  force  at  /is  sus¬ 
tained  by  G  /and  /  H,  the  former  stretched  and  the  latter  com¬ 
pressed.  The  piece,  C  D ,  is  neither  stretched  nor  pressed,  and 
could  be  dispensed  with  if  the  joinings  at  /and  1  could  be  made 
as  effectually  without  it.  In  case  A  B  should  fail,  then  C  D 
would  be  in  a  state  of  tension. 

257. —  The  pressure  of  inclined  beams.  The  centre  of  gravi¬ 
ty  of  a  uniform  prism  or  cylinder,  is  in  its  axis,  at  the  middle  of 
its  length.  In  irregular  bodies  with  plain  sides,  the  centre  of 


140 


AMERICAN  HOUSE-CARPENTER. 


gravity  may  be  found  by  balancing  them  upon  the  edge  of  a  prism 
in  two  positions,  making  a  line  each  time  upon  the  body  in  a  line 
with  the  edge  of  the  prism,  and  the  intersection  of  those  lines 
will  indicate  the  point  required, 


An  inclined  post  or  strut,  supporting  some  heavy  pressure  ap¬ 
plied  at  its  upper  end,  as  at  Fig.  186,  exerts  a  pressure  at  its  foot 
in  the  direction  of  its  length,  or  nearly  so.  But  when  such  a 
beam  is  loaded  uniformly  over  its  whole  length,  as  the  rafter  of  a 
roof,  the  pressure  at  its  foot  varies  considerably  from  the  direction 
of  its  length.  For  example,  let  A  B ,  {Fig.  193,)  be  a  beam  lean¬ 
ing  against  the  wall,  B  c,  and  supported  at  its  foot  by  the  abut¬ 
ment,  A ,  in  the  beam,  A  c,  and  let  o  be  the  centre  of  gravity  of  the 
beam.  Through  o,  draw  the  vertical  line,  b  d,  and  from  B,  draw 
the  horizontal  line,  B  b,  cutting  b  d  in  b;  join  b  and  A,  and  b  A 
will  be  the  direction  of  the  thrust.  To  prevent  the  beam  from 
loosing  its  footing,  the  joint  at  A  should  be  made  at  right  angles 
to  b  A.  The  amount  of  pressure  will  bo  found  thus :  let  b  d, 
(by  any  scale  of  equal  parts,)  equal  the  number  of  tons,  cwts., 
or  pounds  weight  upon  the  beam,  A  B  ;  draw  d  e ,  parallel  to  B 
b  ;  then  b  e,  (by  the  same  scale,)  equals  the  pressure  in  the  direc¬ 
tion,  b  A  ;  and  e  d ,  the  pressure  against  the  wall  at  B — and  also 
the  horizontal  thrust  at  A,  as  these  are  always  equal  in  a  construc¬ 
tion  of  this  kind.  Fig.  194  represents  two  equal  beams,  sup¬ 
ported  at  their  feet  by  the  abutments  in  the  tie-beam.  This  case 
is  similar  to  the  last ;  for  it  is  obvious  that  each  beam  is  in  pre¬ 
cisely  the  position  of  the  beam  in  Fig.  193.  The  horizontal 


FRAMING. 


141 


B 


pressures  at  B,  being  equal  and  opposite,  balance  one  another ; 
and  their  horizontal  thrusts  at  the  tie-beam  are  also  equal.  (See 
Art.  250 — Fig.  186.)  When  the  inclination  of  a  roof,  (Fig. 
194,)  is  one-fourth  of  the  span,  or  of  ashed,  (Fig.  193,)  is  one-half 
the  span,  the  horizontal  thrust  of  a  rafter,  whose  centre  of  gravity 
is  at  the  middle  of  its  length,  is  exactly  equal  to  the  weight  dis¬ 
tributed  uniformly  over  its  surface.  The  inclination,  in  a  rafter 
uniformly  loaded,  which  will  produce  the  least  oblique  pressure, 
( b  e ,  Fig.  193,)  is  35  degrees  and  16  minutes. 


258.  — In  shed,  or  lean-to  roofs,  as  Fig.  193,  the  horizontal 
pressure  will  be  entirely  removed,  if  the  bearings  of  the  rafters,  as 
A,  B,  (Fig.  195,)  are  made  horizontal — provided,  however,  that 
the  rafters  and  other  framing  do  not  bend  between  the  points  of 
support.  If  a  beam  or  rafter  have  a  natural  curve,  the  convex 
or  rounding  edge  should  be  laid  uppermost. 

259.  — A  beam  laid  horizontally,  supported  at  each  end  and 
uniformly  loaded,  is  subject  to  the  greatest  strain  at  the  middle 


142 


AMERICAN  HOUSE-CARPENTER. 


of  its  length.  The  amount  of  pressure  at  that  point  is  equal  to 
half  of  the  whole  load  sustained.  The  greatest  strain  coming 
upon  the  middle  of  such  a  beam,  mortices,  large  knots  and  other 
defects,  should  be  kept  as  far  as  possible  from  that  point ;  and,  in 
resting  a  load  upon  a  beam,  as  a  partition  upon  a  floor  beam,  the 
weight  should  be  so  adjusted  that  it  will  bear  at  or  near  the  ends. 
{See  Art.  2S2.) 

260. —  The  resistance  of  timber.  When  the  stress  that  a 
given  load  exerts  in  any  particular  direction,  has  been  ascertain¬ 
ed,  before  the  proper  size  of  the  timber  can  be  determined  for  the 
resistance  of  that  pressure,  the  strength  of  the  kind  of  timber  to 
be  used  must  be  known.  The  following  rules  for  calculating  the 
resistance  of  timber,  are  based  upon  the  supposition  that  the  tim¬ 
ber  used  be  of  what  is  called  “  merchantable”  quality — that  is, 
strait-grained,  seasoned,  and  free  from  large  knots,  splits,  decay, 
&c. 


Fig.  198. 


The  strength  of  a  piece  of  timber,  is  to  be  considered  in  ac¬ 
cordance  with  the  direction  in  which  the  strain  is  applied  upon 


FRAMING. 


143 


It.  When  it  is  compressed  in  the  direction  of  its  length,  as  in 
Fig.  196,  its  strength  is  termed  the  resistance  to  compression. 
When  the  force  tends  to  pull  it  asunder  in  the  direction  of  its 
length,  (A,  Fig.  197,)  it  is  termed  the  resistance  to  tension. 
And  when  strained  by  a  force  tending  to  break  it  crosswise,  as  at 
Fig.  198,  its  strength  is  called  the  resistance  to  cross  strains. 

261. — Resistance  to  compression.  When  the  height  of  a 
piece  of  timber  exceeds  about  10  times  its  diameter  if  round,  or 
10  times  its  thickness  if  rectangular,  it  will  bend  before  crushing. 
The  first  of  the  following  cases,  therefore,  refers  to  such  posts  as 
would  be  crushed  if  overloaded,  and  the  other  two  to  such  as 
would  bend  before  crushing.  In  estimating  the  strength  of  tim¬ 
ber  for  this  kind  of  resistance,  it  is  provided  in  the  following 
rules  that  the  pressure  be  exactly  in  a  line  with  the  axis  of  the 
post. 

Case  1. — To  find  the  area  of  a  post  that  will  safely  bear  a  given 
weight — when  the  height  of  the  post  is  less  than  10  times  its  least 
thickness.  Rule. — Divide  the  given  weight  in  pounds  by  1000 
for  pine  and  1400  for  oak,  and  the  quotient  will  be  the  least  area 
of  the  post  in  inches.  This  rule  requires  that  the  area  of  the 
abutting  surface  be  equal  to  the  result :  should  there  be,  there¬ 
fore,  a  tenon  on  the  end  of  the  post,  this  quotient  will  be  too  small. 
Example. — What  should  be  the  least  area  of  a  pine  post  that  will 
safely  sustain  48,000  pounds  ?  48,000,  divided  by  1000,  gives 

48 — the  required  area  in  inches.  Such  a  post  may  be  6x8 
inches,  and  will  bear  to  be  of  any  length  within  10  times  6  inches, 
its  least  thickness. 

Case  2. — To  find  the  area  of  a  rectangular  post  that  will 
safely  bear  a  given  weight — when  its  height  is  10  times  its  least 
thickness  or  more.  Rule . — Multiply  the  given  weight  or  pres¬ 
sure  in  pounds  by  the  square  of  the  length  in  feet ;  and  multi¬ 
ply  this  product  by  the  decimal,  '0015,  for  oak,  -0021,  for  pitch 
pine  and  ‘0016  for  white  pine  ;  then  divide  this  product  by  the 
breadth  in  inches,  and  the  cube-root  of  the  quotient  will  be  the 


144 


AMERICAN  HOUSE-CARPENTER. 


thickness  in  inches.  Example. — What  should  be  the  thickness 
of  a  pine  post,  8  feet  high  and  8  inches  wide,  in  order  to  support 
a  weight  of  12  tons,  or  26,880  pounds  ?  The  square  of  the  length 
is  64  feet;  this,  multiplied  by  the  weight  in  pounds,  gives 
1,730,320;  this  product,  multiplied  by  the  decimal,  ’0016,  gives 
2768-512  ;  and  this  again,  divided  by  the  breadth  in  inches,  gives 
346-064 ;  by  reference  to  the  table  of  cube-roots  in  the  appendix, 
the  cube-root  of  this  number  will  be  found  to  be  7  inches  large — • 
which  is  the  thickness  required.  The  stiffest  rectangular  post  is 
that  in  which  the  sides  are  as  10  to  6. 

Case  3.— To  find  the  area  of  a.  round,  or  cylindrical,  post,  that 
will  safely  bear  a  given  weight— -when  its  height  is  10  times  its 
least  diameter  or  more.  Rule—  Multiply  the  given  weight  or 
pressure  in  pounds  by  1-7,  and  the  product  by  -0015  for  oak,  -0021 
for  pitch  pine  and  -0016  for  white  pine  ;  then  multiply  the  square- 
root  of  this  product  by  the  height  in  feet,  and  the  square-root  of 
the  last  product  will  be  the  diameter  required,  in  inches.  Exam¬ 
ple. — What  should  be  the  diameter  of  a  cylindrical  oak  post,  8 
feet  high,  in  order  to  support  a  weight  of  12  tons,  Or  26,880 
pounds  ?  This  weight  in  pounds,  multiplied  by  1-7,  gives  45,696 ; 
and  this,  by  -0015,  gives  68-544  ;  the  square-root  of  this  product 
is  (by  the  table  in  the  appendix)  8‘28,  nearly — which,  multiplied 
by  8,  gives  66-24  ;  the  square-root  of  this  number  is  8*14,  nearly  ; 
therefore,  8-14  inches  is  the  diameter  required. 

Experiments  have  shown  that  the  pressure  should  never  be 
more  than  1000  pounds  per  square  inch  on  a  joint  in  yellow  pine 
— when  the  end  of  the  grain  of  one  piece  is  pressed  against  the 
side  of  the  grain  of  the  other. 

262. — Resistance  to  tension.  A  bar  of  oak  of  an  inch  square, 
pulled  in  the  direction  of  its  length,  has  been  tom  asunder  by  a 
weight  of  11,500  lbs. 

Of  white  pine  -  11,000 

Of  pitch  pine  -  10,000 


/ 


FRAMING.  145 

Therefore,'  when  the  strain  is  applied  in  a  line  with  the  axis  of 
the  piece,  the  following  rule  must  be  observed. 

To  find  the  area  of  a  piece  of  timber  to  resist  a  given  strain  in 
the  direction  of  its  length.  Rule. — Divide  the  given  weight  to 
be  sustained,  by  the  weight  that  will  tear  asunder  a  bar  an  inch 
square  of  the  same  kind  of  wood,  (as  above.)  and  the  product  will 
be  the  area  in  inches  of  a  piece  that  will  just  sustain  the  given 
weight ;  but  the  area  should  be  at  least  4  times  this,  to  safely 
sustain  a  constant  load  of  the  given  weight.  Example. — What 
should  be  the  area  of  a  stick  of  pitch  pine  timber,  which  is  re¬ 
quired  to  sustain  safely  a  constant  load  of  60,000  pounds  ?  60,000, 

divided  by  10,000,  (as  above,)  gives  6,  and  this,  multiplied  by  4, 
give  24  inches — the  answer. 

263. — Resistance  to  cross  strains.  To  find  the  scantling  of  a 
piece  of  timber  to  sustain  a  given  weight,  when  such  piece  is 
supported  at  the  ends  in  a  horizontal  position. 

Case  1. — When  the  breadth  is  given.  Rule. — Multiply  the 
square  of  the  length  in  feet  by  the  weight  in  pounds,  and  this 
product  by  the  decimal,  *009,  for  oak,  *011  for  white  pine  and  *016 
for  pitch  pine  ;  divide  the  product  by  the  breadth  in  inches,  and 
the  cube-root  of  the  quotient  will  be  the  depth  required  in  inches. 
Examplei — What  should  be  the  depth  of  a  beam  of  white  pine, 
having  a  bearing  of  24  feet  and  a  breadth  of  6  inches,  in  order  to 
support  900  pounds  ?  The  square  of  24  is  57 6,  and  this,  multiplied 
by  900,  gives  518-400;  and  this  again,  by  'Oil,  gives  5702*400  ; 
this,  divided  by  6,  gives  950*400  ;  the  cube-root  of  which  is  9-83 
inches — the  depth  required. 

Case  2. — When  the  depth  is  given.  Rule. — Multiply  the 
square  of  the  length  in  feet  by  the  weight  in  pounds,  and  multi¬ 
ply  this  product  by  the  decimal,  *009,  for  oak,  *011  for  white  pine 
and  *016  for  pitch  pine ;  divide  the  last  product  by  the  cube  of 
the  depth  in  inches,  and  the  quotient  will  be  the  breadth  in  inches 
required.  Example. — What  should  be  the  breadth  of  a  beam  of 
oak,  having  a  bearing  of  16  feet  and  a  depth  of  12  inches*  in 

19 


146 


AMERICAN  HOUSE-CARPENTER. 


order  to  support  a  weight  of  4000  pounds?  The  square  of  16  is 
256,  which,  multiplied  by  4000,  gives  1,024,000  ;  this,  multiplied 
by  -009,  gives  9216  ;  and  this  again,  divided  by  1728,  the  cube  of 
12,  gives  5£  inches — which  is  the  breadth  required. 

Case  3. — When  the  breadth  bears  a  certain  proportion  to  the 
depth.  When  neither  the  breadth  nor  depth  is  given,  it  will  be 
best  to  fix  on  some  proportion  which  the  breadth  should  have  to 
the  depth ;  for  instance,  suppose  it  be  convenient  to  make  the 
breadth  to  the  depth  as  0 ‘6  is  to  1,  then  the  rule  would  become  as 
follows  :  Rule. — Multiply  the  weight  in  pounds  by  the  decimal, 
•009,  for  oak,  ‘Oil  for  white  pine  and  ’016  for  pitch  pine;  divide 
the  product  by  0*6,  and  extract  the  square-root ;  multiply  this  root 
by  the  length  in  feet,  and  extract  the  square-root  a  second  time, 
which  will  be  the  depth  in  inches  required.  The  breadth  is 
equal  to  the  depth  multiplied  by  the  decimal,  0-6.  It  is  obvious 
that  any  other  proportion  of  the  breadth  and  depth  may  be  ob¬ 
tained  by  merely  changing  the  decimal,  0-6,  in  the  rule.  Exam¬ 
ple. — What  should  be  the  depth  and  breadth  of  a  beam  of  pitch 
pine,  having  a  proportion  to  one  another  as  0  6  to  1,  and  a  bearing 
of  22  feet,  in  order  to  sustain  a  ton  weight,  or  2240  pounds  ? 
This,  multiplied  by  *016,  gives  35-84,  which,  divided  by  0‘6, 
gives  59-73  ;  the  square-root  of  this  is  7-7,  which,  multiplied  by 
22,  the  length,  gives  169-4;  the  square-root  of  this  is  13 — which 
is  the  depth  required.  Then  13,  multiplied  by  0-6,  gives  7-8 
inches — the  required  breadth. 

Case  4. — When  the  beam  is  inclined,  as  A  B,  Fig.  193. 
Rule. — Multiply  together  the  weight  in  pounds,  the  length  of  the 
beam  in  feet,  the  horizontal  distance,  A  c,  between  the  supports, 
in  feet,  and  the  decimal,  -009,  for  oak,  -Oil  for  white  pine,  and 
•016  for  pitch  pine  ;  divide  this  product  by  0-6,  and  the  fourth 
root  of  the  quotient  will  give  the  depth  in  inches.  The  breadth 
is  equal  to  the  depth  multiplied  by  the  decimal,  0-6.  Example. — 
What  should  be  the  size  of  an  oak  beam,  the  sides  to  bear  a  pro¬ 
portion  to  one  another  as  0-6  to  1,  in  order  to  support  a  ton  weight 


FRAMING. 


147 


or  2240  pounds,  the  beam  being  inclined  so  that,  its  length  being 
20  feet,  its  horizontal  distance  between  the  points  of  support  will 
be  16  feet?  2240,  multiplied  by  20,  gives  44,800,  which,  multi¬ 
plied  by  16,  gives  716,800  ;  and  this  again,  by  the  decimal,  -009, 
gives  6451-2 ;  this  last,  divided  by  06,  gives  10,752,  the  fourth 
root  of  which  is  10-18,  nearly  ;  and  this,  multiplied  by  0‘6,  gives 
6-1 ;  therefore,  the  size  of  the  beam  should  be  10-18  inches  by 
6-1  inches. 


264.  —  To  ascertain  the  scantling  of  the  stiffest  beam  that 
can  be  cut  from  a  cylinder.  Let  d  a  c  b,  (Fig.  199,)  be  the  sec¬ 
tion,  and  e  the  centre,  of  a  given  cylinder.  Draw  the  diameter, 
a  b  ;  upon  a  and  6,  with  the  radius  of  the  section,  describe  the 
arcs,  d  e  and  e  c  ;  join  d  and  a,  a  and  c,  c  and  b,  and  b  and  d  ; 
then  the  rectangle,  d  a  cb,  will  be  a  section  of  the  beam  required. 

265.  — The  greater  the  depth  of  a  beam  in  proportion  to  the 
thickness,  the  greater  the  strength.  Bat  when  the  difference  be¬ 
tween  the  depth  and  the  breadth  is  great,  the  beam  must  be 
stayed,  (as  at  Fig.  202,)  to  prevent  its  falling  over  and  breaking 
sideways.  Their  shrinking  is  another  objection  to  deep  beams  ; 
but  where  these  evils  can  be  remedied,  the  advantage  of  increas¬ 
ing  the  depth  is  considerable.  The  following  rule  is,  to  find  the 
strongest  form  for  abeam  out  of  a  given  quantity  of  timber. 
Rule. — Multiply  the  length  in  feet  by  the  decimal,  0-6,  and  divide 
the  given  area  in  inches  by  the  product ;  and  the  square  of  the 
quotient  will  give  the  depth  in  inches.  Example. — What  is  the 
strongest  form  for  a  beam  whose  given  area  of  section  is  48 


148 


AMERICAN  HOUSE-CARPENTER. 


inches,  and  length  of  bearing  20  feet  ?  The  length  in  feet,  20, 
multiplied  by  the  decimal,  0  6,  gives  12;  the  given  area  in  inches, 
48,  divided  by  12,  gives  a  quotient  of  4,  the  square  of  which  is 
16 — this  is  the  depth  in  inches ;  and  the  breadth  must  be  3 
inches.  A  beam  16  inches  by  3  would  hear  twice  as  much  as  a 
square  beam  of  the  same  area  of  section;  which  shows  how  im¬ 
portant  it  is  to  make  beams  deep  and  thin.  In  many  old  build¬ 
ings,  and  even  in  new  ones,  in  country  places,  the  very  reverse  of 
this  has  been  practised ;  the  principal  beams  being  oftener  laid 
on  the  broad  side  than  on  the  narrower  one. 

266.  — Systems  of  Framing.  In  the  various  parts  of  framing 
known  as  floors,  partitions,  roofs,  bridges,  &c,,  each  has  a  specific 
object ;  and,  in  all  designs  for  such  constructions,  this  object 
should  be  kept  clearly  in  view  ;  the  various  parts  being  so  dis¬ 
posed  as  to  serve  the  design  with  the  least  quantity  of  material. 
The  simplest  form  is  the  best,  not  only  because  it  is  the  most 
economical,  but  for  many  other  reasons.  The  great  number  of 
joints,  in  a  complex  design,  render  the  construction  liable  to  de¬ 
rangement  by  multiplied  compressions,  shrinkage,  and,  in  conse¬ 
quence,  highly  increased  oblique  strains ;  by  which  its  stability 
and  durability  are  greatly  lessened. 

FLOORS. 

267. — -Floors  have  been  constructed  in  various  ways,  and  are 
known  as  single-joisted ,  double ,  and  framed.  In  a  single- 
joisted  floor,  the  timbers,  or  floor-joists,  are  disposed  as  is  shown  in 
Fig.  200.  Where  strength  is  the  principal  object,  this  manner 
of  disposing  the  floor-joists  is  far  preferable  ;  as  experiments  have 
proved  that,  with  the  same  quantity  of  material,  single-joisted 
floors  are  much  stronger  than  either  double  or  framed  floors. 
To  obtain  the  greatest  strength,  the  joists  should  be  thin  and 
deep. 

268.  —  To  find  the  depth  of  a  joist,  the  length  of  bearing 
and  thickness  being  given ,  when  the  distance  from  centres  is 


FRAMING. 


149 


Fig.  200. 


12  inches.  Rule. — Divide  the  square  of  the  length  in  feet,  by 
the  breadth  in  inches ;  and  the  cube-root  of  the  quotient,  multi¬ 
plied  by  2-2  for  pine,  or  23  for  oak,  will  give  the  depth  in  inches. 
Example. — What  should  be  the  depth  of  floor-joists,  having  a 
bearing  of  12  feet  and  a  thickness  of  3  inches,  when  said  joists 
are  of  pine  and  placed  12  inches  from  centres  ?  The  square  of 
12  is  144,  which,  divided  by  3,  gives  48  ;  the  cube-root  of  this 
number  is  3'63,  which,  multiplied  by  2‘2,  gives  7’98G  inches, 
the  depth  required  ;  or  8  inches  will  be  found  near  enough  for 
practice. 

269. — Where  chimneys,  flues,  stairs,  &c.,  occur  to  interrupt 
the  bearing,  the  joists  are  framed  into  a  piece,  ( b ,  Fig.  201,) 
called  a  trimmer.  The  beams,  a,  a,  into  which  the  trimmer  is 
framed,  are  called  trimming-beams ,  trimming -joists ,  or  car¬ 
riage-beams.  They  need  to  be  stronger  than  the  common  joists, 
in  proportion  to  the  number  of  beams,  c,  c,  which  they  support. 
The  trimmers  have  to  be  made  strong  enough  to  support  half  the 
weight  which  the  joists,  c,  c,  support,  (the  wall,  or  another  trim¬ 
mer,  at  the  other  end  supporting  the  other  half,)  and  the  carriage- 


150 


AMERICAN  HOUSE-CARPENTER. 


beams  must  each  be  strong  enough  to  support  half  the  weight 
which  the  trimmer  supports.  In  calculating  for  the  dimensions 
of  floor-timbers,  regard  must  he  had  to  the  fact  that  the  weight 
which  they  generally  support — such  as  persons  of  150  pounds 
moving  over  the  floor — exerts  a  much  greater  influence  than 
equal  weights  at  rest.  When  the  trimmer,  b,  is  not  more  dis¬ 
tant  from  the  bearing,  d ,  than  is  necessary  for  ordinary  hearths, 
&c.,  it  will  be  sufficient  to  add  £  of  an  inch  to  the  thickness  of 
the  carriage-beam  for  every  joist,  c,  that  is  supported.  Thus,  if 
the  thickness  of  c  is  3  inches,  and  the  number  of  joists  supported 
be  6,  add  6  eighths,  or  f  of  an  inch,  making  the  carriage-beams 
3f  inches  thick.  It  is  generally  the  practice  in  dwellings  to  make 
the  carriage-beam,  in  all  situations,  one  inch  thicker  than  the 
common  joists.  But  it  is  well  to  have  a  rule  for  determining  the 
size  more  accurately  in  extreme  cases. 

270. — When  the  bearing  exceeds  8  feet,  there  should  be  struts, 
as  a  and  a,  {Fig.  202,)  well  nailed  between  the  joists.  These 
will  prevent  the  turning  or  twisting  of  the  floor-joists,  and  will 
greatly  stiffen  the  floor.  For,  in  the  event  of  a  heavy  weight 
resting  upon  one  of  the  joists,  these  struts  will  prevent  that  joist 
from  settling  below  the  others,  to  the  injury  of  the  plastering 


FRAMING. 


151 


Fig.  202. 


upon  the  underside.  When  the  length  of  bearing  is  great,  struts 
should  be  inserted  at  about  every  4  feet. 

271.  — Single-joisted  floors  may  be  constructed  for  as  great  a 
length  of  bearing  as  timber  of  sufficient  depth  can  be  obtained  ; 
but,  in  such  cases,  where  perfect  ceilings  are  desirable,  either 
double  or  framed  floors  are  considered  necessary.  Yet  the  ceil¬ 
ings  under  a  single-joisted  floor  may  be  rendered  more  durable  by 
cross-furring ,  as  it  is  termed — which  consists  of  nailing  a  series 
of  narrow  strips  of  board  on  the  under  edge  of  the  beams  and  at 
right  angles  to  them.  To  these,  instead  of  the  beams,  the  laths 
are  nailed.  The  strips  should  be  not  over  2  inches  wide — enough 
to  join  the  laths  upon  is  all  that  is  wanted  in  width — and  not 
more  than  12  inches  apart.  It  is  necessary  that  all  furring  for 
plastering  be  narrow,  in  order  that  the  mortar  may  have  a  suffi¬ 
cient  clinch. 

When  it  is  desirable  to  prevent  the  passage  of  sound,  the  open¬ 
ings  between  the  beams,  at  about  3  inches  from  the  upper  edge, 
are  closed  by  short  pieces  of  boards,  which  rest  on  cleets  nailed 
to  the  beam  along  its  whole  length.  This  forms  a  floor  upon 
which  mortar  is  laid  to  the  depth  of  about  2  inches,  leaving  but 
about  half  an  inch  from  its  upper  surface  to  the  under  side  of  the 
floor-plank. 

272.  — Double  floors.  A  double  floor  consists,  as  at  Fig.  203, 
of  three  tiers  of  joists  or  timbers ;  viz.,  bridging-joists ,  a ,  a, 
binding-joists ,  b,  b,  and  ceiling-joists ,  c,  c.  The  binding-joists 


Fig.  203. 


are  the  principal  support,  and  of  course  reach  from  wall  to  wall. 
The  bridging-joists,  which  support  the  floor-plank,  are  laid  upon 
the  binding-joists,  to  which  they  are  nailed;  sometimes  they  are 
notched  into  the  binding-joists,  but  they  are  sufficiently  firm 
when  well  nailed.  The  ceiling-joists  are  notched  into  the  under 
side  of  the  binders,  and  nailed ;  they  are  the  support  of  the  lath 
and  plastering. 

273. — Binders  are  laid  6  feet  apart.  At  this  distance  the  fol¬ 
lowing  rules  will  give  the  scantling. 

Case  1. — To  find  the  depth  of  a  binding-joist,  the  length  and 
breadth  being  given.  Rule. — Divide  the  square  of  the  length  in 
feet,  by  the  breadth  in  inches  ;  and  the  cube-root  of  the  quotient, 
multiplied  by  3-42  for  pine,  or  by  353  for  oak,  will  give  the  depth 
in  inches.  Example. — What  should  be  the  depth  of  a  binding- 
joist,  having  a  length  of  12  feet  and  a  breadth  of  6  inches,  when 
the  kind  of  timber  is  pine  ?  The  square  of  12  is  144,  which,  di¬ 
vided  by  6,  gives  24  ;  the  cube-root  of  this  is  2-88,  which,  multi¬ 
plied  by  3-42,  gives  9-85,  the  depth  in  inches. 

Case  2. — To  find  the  breadth,  when  the  depth  and  length  are 
given.  Rule. — Divide  the  square  of  the  length  in  feet,  by  the 


FRAMING. 


153 


cube  of  the  depth  in  inches  ;  and  multiply  the  quotient  by  40  for 
pine,  or  by  44  for  oak,  which  will  give  the  breadth  in  inches. 
Example. — What  should  be  the  breadth  of  a  binding-joist,  hav¬ 
ing  a  length  of  12  feet  and  a  depth  of  10  inches,  when  the  kind 
of  wood  is  pine  1  The  cube  of  10  is  1000  ;  the  square  of  12  is 
144;  this,  divided  by  1000,  gives  a  quotient  of  T44;  and  this 
quotient,  multiplied  by  40,  gives  5‘76,  the  breadth  in  inches. 

274.  — Bridging-joists  are  laid  from  12  to  20  inches  apart.  The 
scantling  may  be  found  by  the  rule  at  Art.  268. 

275.  — Ceiling-joists  are  generally  placed  12  inches  apart  from 
centres.  They  are  arranged  to  suit  the  length  of  the  lath  ;  this 
being,  in  most  cases,  4  feet  long.  What  is  said  at  Art.  271,  in 
regard  to  the  width  of  furring  for  plastering,  will  apply  to  the 
thickness  of  ceiling-joists. 

To  find  the  depth  of  a  ceiling-joist,  when  the  length  of  bearing 
and  thickness  are  given.  Rule. — Divide  the  length  in  feet  by 
the  cube-root  of  the  breadth  in  inches ;  and  multiply  the  quotient 
by  064  for  pine,  or  by  0*67  for  oak,  which  will  give  the  depth  in 
inches.  Example. — What  should  be  the  depth  of  a  ceiling-joist 
of  pine,  when  the  length  of  bearing  is  G  feet  and  the  thickness  2 
inches  ?  The  length  in  feet,  6,  divided  by  the  cube-root  of  the 
breadth  in  inches,  1*26,  gives  a  quotient  of  4‘76,  which,  being 
multiplied  by  the  decimal,  0-64,  gives  3  inches,  the  depth  re¬ 
quired. 

When  the  thickness  of  a  ceiling-joist  is  2  inches,  the  depth  in 
inches  will  be  equal  to  half  the  length  of  bearing  in  feet.  Thus, 
if  the  bearing  is  6  feet,  the  depth  will  be  3  inches  ;  bearing  8 
feet,  depth  4  inches,  &c. 

276.  — Framed  floors.  When  a  good  ceiling  is  required,  and 
the  distance  of  bearing  is  great,  the  binding-joists,  instead  of 
reaching  from  wall  to  wall,  are  framed  into  girders.  These  are 
heavy  timbers,  as  d,  (Fig.  204,)  which  reach  from  wall  to  wall, 
being  the  chief  support  of  the  floor.  Such  an  arrangement  is 
termed  a  framed  floor.  The  binding,  the  bridging  and  the  ceil- 

20 


154  AMERICAN  HOUSE-CARPENTER. 


ing-joists  in  these,  are  the  same  as  those  in  double  floors  just 
described.  The  distinctive  feature  of  this  kind  of  floor  is  the 
girder. 

277. — Girders  should  be  made  as  deep  as  the  timber  will  allow : 
if  their  being  increased  in  size  should  reduce  the  height  of  a  story 
a  few  inches,  it  would  be  better  than  to  have  a  house  suffer  from 
defective  ceilings  and  insecure  floors.  In  the  following  rules  for 
the  scantling  of  girders,  they  are  supposed  to  be  placed  at  10  feet 
apart. 

Case  1. — To  find  the  depth,  when  the  breadth  of  the  girder 
and  the  length  of  bearing  are  given.  Rule. — Divide  the  square 
of  the  length  in  feet,  by  the  breadth  in  inches;  and  the  cube-root 
of  the  quotient,  multiplied  by  4’2  for  pine,  or  by  43  for  oak,  will 
give  the  depth  required  in  inches.  Example. — What  should  be 
the  depth  of  a  pine  girder,  having  a  length  of  20  feet  and  a  breadth 
of  13  inches  ?  The  square  of  20  is  400,  which,  divided  by  13, 
gives  30'77 ;  the  cube-root  of  this  is  3-12,  which,  multiplied  by 
4‘2,  gives  13  inches,  the  depth  required. 


FRAMING. 


155 


Case  2. — To  find  the  breadth,  when  the  length  of  bearing  and 
depth  are  given.  Rule. — Divide  the  square  of  the  length  in  feet, 
by  the  cube  of  the  depth  in  inches ;  and  the  quotient,  multiplied 
by  74  for  pine,  or  by  82  for  oak,  will  give  the  breadth  in  inches. 
Example. — What  should  be  the  breadth  of  a  pine  girder,  having 
a  length  of  18  feet  and  a  depth  of  14  inches  ?  The  square  of 
the  length  in  feet,  324,  divided  by  the  cube  of  the  depth  in 
inches,  2744,  gives  T18  ;  and  this,  multiplied  by  74,  gives  8‘73 
inches,  the  breadth  required. 

278. — When  the  breadth  of  a  girder  is  more  than  about  12 
inches,  it  is  recommended  to  divide  it  by  sawing  from  end  to  end, 
vertically  through  the  middle,  and  then  to  bolt  it  together  with 
the  sawn  sides  outwards.  This  is  not  to  strengthen  the  girder, 
as  some  have  supposed,  but  to  reduce  the  size  of  the  timber,  in 
order  that  it  may  dry  sooner.  The  operation  affords  also  an  op¬ 
portunity  to  examine  the  heart  of  the  stick — a  necessary  precau¬ 
tion  ;  as  large  trees  are  frequently  in  a  state  of  decay  at  the  heart, 
although  outwardly  they  are  seemingly  sound.  When  the  halves 
are  bolted  together,  thin  slips  of  wood  should  be  inserted  between 
them  at  the  several  points  at  which  they  are  bolted,  in  order  to 
leave  sufficient  space  for  the  air  to  circulate  between.  This 
tends  to  prevent  decay  ;  which  will  be  found  first  at  such  parts 
as  are  not  exactly  tight,  nor  yet  far  enough  apart  to  permit  the 
escape  of  moisture. 

271b — When  girders  are  required  for  a  long  bearing,  it  is  usual 
to  truss  them  ;  that  is,  to  insert  between  the  halves  two  pieces  of 
oak  which  are  inclined  towards  each  other,  and  which  meet  at 
the  centre  of  the  length  of  the  girder,  like  the  rafters  of  a  roof- 
truss,  though  nearly  if  not  quite  concealed  within  the  girder. 
This,  and  many  similar  methods,  though  extensively  practised, 
are  generally  worse  than  useless  ;  since  it  has  been  ascertained 
that,  in  nearly  all  such  cases,  the  operation  has  positively  weak¬ 
ened  the  girder. 

A  girder  may  be  strengthened  by  mechanical  contrivance,  when 


156 


AMERICAN  HOUSE-CARPENTER. 


its  depth  is  required  to  be  greater  than  any  one  piece  of  timber 
will  allow.  Fig.  205  shows  a  very  simple  yet  scientific  method 
of  doing  this.  The  two  pieces  of  which  the  girder  is  composed 
are  bolted,  or  pinned,  together,  having  keys  inserted  between  to 
prevent  the  pieces  from  sliding.  The  keys  should  be  of  hard 
wood,  well  seasoned.  The  two  pieces  should  be  about  equal  in 
depth,  in  order  that  the  joint  between  them  may  be  in  the  neutral 
line.  (See  Art.  254.)  The  thickness  of  the  keys  should  be 
about  half  their  breadth,  and  the  amount  of  their  united  thick¬ 
nesses  should  be  equal  to  a  trifle  over  the  depth  and  one-third  of 
the  depth  of  the  girder.  Instead  of  bolts  or  pins,  iron  hoops  are 
sometimes  used ;  and  when  they  can  be  procured,  they  are  far 
preferable.  In  this  case,  the  girder  is  diminished  at  the  ends, 
and  the  hoops  driven  from  each  end  towards  the  middle. 

280. — Beams  may  be  spliced,  if  none  of  a  sufficient  length  can 
be  obtained,  though  not  at  or  near  the  middle,  if  it  can  be  avoided. 
(See  Art.  259  and  332.)  Girders  should  rest  from  9  to  12  inches  on 
the  wall,  and  a  space  should  be  left  for  the  air  to  circulate  around 
the  ends,  that  the  dampness  may  evaporate.  Floor-timbers  are 
supported  at  their  ends  by  walls  of  considerable  height.  They 
should  not  be  permitted  to  rest  upon  intervening  partitions,  which 
are  not  likely  to  settle  as  much  as  the  walls  ;  otherwise  the  une¬ 
qual  settlements  will  derange  the  level  of  the  floor.  As  all  floors, 
however  well-constructed,  settle  in  some  degree,  it  is  advisable  to 


FRAMING. 


157 


frame  the  joists  a  little  higher  at  the  middle  of  the  room  than  at 
its  sides, — as  also  the  ceiling-joists  and  cross-furring,  when  either 
are  used.  In  single-joisted  floors,  for  the  same  reason,  the 
rounded  edge  of  the  stick,  if  it  have  one,  should  be  placed  up¬ 
permost. 

If  the  floor-plank  are  laid  down  temporarily  at  first,  and  left  to 
season  a  few  months  before  they  are  finally  driven  together  and 
secured,  the  joints  will  remain  much  closer.  But  if  the  edges  of 
the  plank  are  planed  after  the  first  laying,  they  will  shrink  again  ; 
as  it  is  the  nature  of  wood  to  shrink  after  every  planing  however 
dry  it  may  have  been  before. 

PARTITIONS. 

% 

281.  — Too  little  attention  has  been  given  to  the  construction  of 
this  part  of  the  frame- work  of  a  house.  The  settling  of  floors 
and  the  cracking  of  ceilings  and  walls,  which  disfigure  to  so  great 
an  extent  the  apartments  of  even  our  most  costly  houses,  may  be 
attributed  almost  solely  to  this  negligence.  A  square  of  parti¬ 
tioning  weighs  about  half  a  ton,  a  greater  weight,  when 
added  to  its  customary  load,  such  as  furniture,  storage, 
&c.,  than  any  ordinary  floor  is  calculated  to  sustain.  Hence 
the  timbers  bend,  the  ceilings  and  cornices  crack,  and  the  whole 
interior  part  of  the  house  settles ;  showing  the  necessity  for 
providing  adequate  supports  independent  of  the  floor-timbers. 
A  partition  should,  if  practicable,  be  supported  by  the  walls 
with  which  it  is  connected,  in  order,  if  the  walls  settle,  that 
it  may  settle  with  them.  This  would  prevent  the  separation  of 
the  plastering  at  the  angles  of  rooms.  For  the  same  reason,  a 
firm  connection  with  the  ceiling  is  an  important  object  in  the  con¬ 
struction  of  a  partition. 

282.  — The  joists  in  a  partition  should  be  so  placed  as  to  dis¬ 
charge  the  weight  upon  the  points  of  support.  All  oblique  pieces 
in  a  partition,  that  tend  not  to  this  object,  are  much  better  omitted. 
Fig.  206  represents  a  partition  having  a  door  in  the  middle.  Its 


158  AMERICAN  HOUSE-CARPENTER. 


construction  is  simple  but  effective.  Fig.  207  shows  the  manner 
of  constructing  a  partition  having  doors  near  the  ends.  The  truss 
is  formed  above  the  door-heads,  and  the  lower  parts  are  suspended 
from  it.  The  posts,  a  and  6,  are  halved,  and  nailed  to  the  tie,  c  d, 
and  the  sill,  ef.  The  braces  in  a  trussed  partition  should  be 
placed  so  as  to  form,  as  near  as  possible,  an  angle  of  40  degrees 
with  the  horizon.  In  partitions  that  are  intended  to  support  only 
their  own  weight,  the  principal  timbers  may  be  3x4  inches  for  a 
20  feet  span,  3|x5  for  30  feet,  and  4x6  for  40.  The  thickness  of 
the  filling-in  stuff  may  be  regulated  according  to  what  is  said  at 
Art.  271,  in  regard  to  the  width  of  furring  for  plastering.  The 


FRAMING. 


159 


filling-in  pieces  should  be  stiffened  at  about  every  three  feet  by 
short  struts  between. 

All  superfluous  timber,  besides  being  an  unnecessary  load  upon 
the  points  of  support,  tends  to  injure  the  stability  of  the  plaster¬ 
ing  ;  for,  as  the  strength  of  the  plastering  depends,  in  a  great  mea¬ 
sure,  upon  its  clinch,  formed  by  pressing  the  mortar  through  the 
space  between  the  laths,  the  narrower  the  surface,  therefore,  upon 
which  the  laths  are  nailed,  the  less  will  be  the  quantity  of  plas¬ 
tering  unclinched,  and  hence  its  greater  security  from  fractures. 
For  this  reason,  the  principal  timbers  of  the  partition  should  have 
their  edges  reduced,  by  chamfering  off  the  corners. 


w 

pn 

L 

! — 

L 

r 

— 

=i 

= 

— 

— 

= 

— 

i 

— 

— 

i 

— 

— 

— 

— 

— 

— 

y/ 

k 

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Fig.  2U8. 


283. — When  the  principal  timbers  of  a  partition  require  to  be 
large  for  the  purpose  of  greater  strength,  it  is  a  good  plan  to  omit 
the  upright  filling-in  pieces,  and  in  their  stead,  to  place  a  few  hori¬ 
zontal  pieces ;  in  order,  upon  these  and  the  principal  timbers,  to 
nail  upright  battens  at  the  proper  distances  for  lathing,  as  in  Fig. 
208.  A  partition  thus  constructed  requires  a  little  more  space 
than  others  ;  but  it  has  the  advantage  of  insuring  greater  stability 
to  the  plastering,  and  also  of  preventing  to  a  good  degree  the  con¬ 
versation  of  one  room  from  being  heard  in  the  other.  When  a 
partition  is  required  to  support,  in  addition  to  its  own  weight,  that 
of  a  floor  or  some  other  burden  resting  upon  it,  the  dimensions  of 


160 


AMERICAN  HOUSE-CARPENTER. 


the  timbers  may  be  ascertained,  by  applying  the  principles  which 
regulate  the  laws  of  pressure  and  those  of  the  resistance  of  tim¬ 
ber,  as  explained  at  the  first  part  of  this  section.  The  following 
data,  however,  may  assist  in  calculating  the  amount  of  pressure 
upon  partitions : 

284. — The  weight  of  a  square,  (that  is,  a  hundred  square  feet,) 
of  partitioning  may  be  estimated  at  from  1500  to  2000  lbs. ;  a 
square  of  single-joisted  flooring,  at  from  1200  to  2000  lbs. ;  a 
square  of  framed  flooring,  at  from  2700  to  4500  lbs. ;  and  the 
weight  of  a  square  of  deafening ,  (as  described  at  the  latter  part 
of  Art.  271,)  at  about  1500  lbs. 

When  a  floor  is  supported  at  two  opposite  extremities,  and  by  a 
partition  introduced  midway,  one-half  of  the  weight  of  the  whole 
floor  will  then  be  supported  by  the  partition.  As  the  settling  of 
partitions  and  floors,  which  is  so  disastrous  to  plastering,  is  fre¬ 
quently  owing  to  the  shrinking  of  the  timber  and  to  ill-made 
joints,  it  is  very  important  that  the  timber  be  seasoned  and  the 
work  well  executed. 


ROOFS.* 

285.  — In  ancient  buildings,  the  Norman  and  the  Gothic,  the 
walls  and  buttresses  were  erected  so  massive  and  firm,  that  it  was 
customary  to  construct  their  roofs  without  a  tie-beam ;  the  walls 
being  abundantly  capable  of  resisting  the  lateral  pressure  exerted 
by  the  rafters.  But  in  modern  buildings,  the  walls  are  so  slightly 
built  as  to  be  incapable  of  resisting  scarcely  any  oblique  pressure ; 
and  hence  the  necessity  of  constructing  the  roof  so  that  all 
oblique  and  lateral  strains  may  be  removed;  as,  also,  that  instead 
of  having  a  tendency  to  separate  the  walls,  the  roof  may  contri¬ 
bute  to  bind  and  steady  them. 

286.  — In  estimating  the  pressures  upon  any  certain  roof,  for  the 
purpose  of  ascertaining  the  proper  sizes  for  the  timbers,  calcula¬ 
tion  must  be  made  for  the  pressure  exerted  by  the  wind,  and,  if 


*  See  also  Art.  228. 


FRAMING. 


161 


in  a  cold  climate,  for  the  weight  of  snow,  in  addition  to  the  weight 
of  the  materials  of  which  the  roof  is  composed.  The  force  of 
wind  may  be  calculated  at  40  lbs.  on  a  square  foot.  The  weight 
of  snow  will  be  of  course  according  to  the  depth  it  acquires. 
(See  weight  of  materials,  in  Appendix.)  In  a  severe  climate, 
roofs  ought  to  be  constructed  steeper  than  in  a  milder  one ;  in  order 
that  the  snow  may  have  a  tendency  to  slide  off  before  it  becomes  of 
sufficient  weight  to  endanger  the  safety  of  the  roof.  The  inclina^ 
tion  should  be  regulated  in  accordance  with  the  qualities  of  the 
material  with  which  the  roof  is  to  be  covered.  The  following  table 
may  be  useful  in  determining  the  inclination)  and  in  estimating 
the  weight  of  the  various  kinds  of  covering : 


MATERIAL. 

INCLINATION. 

WEIGHT  UPON  A  SQUARE  FOOT. 

Tin, 

Rise  1  inch  to  a  foot. 

f  to  11  lbs. 

Copper, 

<<  i  u  u 

1  to  11  “ 

Lead, 

“  2  inches  “ 

4  to  7  “ 

Zinc, 

a  g  <<  u 

11  to  2  “ 

Short  pine  shingles, 

u  5  a  a 

11  to  21  “ 

Long  cypress  shingles, 

a  g  u  u 

4  to  5  “ 

Slate, 

u  0  u  a 

5  to  9  “ 

The  weight  of  the  covering,  as  above  estimated,  is  that  of  the 
material  only,  added  to  the  weight  of  whatever  is  used  to  fix  it  to 
the  roof,  such  as  nails,  &c. ;  what  the  material  is  laid  on,  such  as 
plank,  boards  or  lath,  is  not  included. 

287. — Fig.  209  to  212  give  a  general  idea  of  the  usual  manner 
of  constructing  trusses  for  roofs:  c,  {Fig.  209,)  is  a  common 


21 


162 


AMERICAN  HOUSE-CARPENTER' 


/ 


FRAMING. 


163 


rafter ;  R  is  a  principal  rafter  ;  k  is  a  king-post ;  5  is  a  stmt ;  /S', 
{Fig.  211,)  is  a  straining-beam  ;  Q  is  a  queen-post ;  T  is  a  tie- 
beam  ;  and  P,  P,  {Fig.  212,)  are  purlins.  In  constructing  a  roof 
of  importance,  the  trusses  should  be  placed  not  over  10  feet  apart, 
the  principal  rafter  supported  by  a  strut  at  every  purlin,  the  purlin 
notched  on  instead  of  being  framed  into  the  principal  rafters,  and 
the  tie-beam  supported  at  proper  distances,  according  to  the  weight 
of  the  ceiling  or  whatever  else  it  is  required  to  support. 

288.  — The  dimensions  of  the  timbers  may  be  found  in  accord¬ 
ance  with  the  principles  explained  at  the  first  part  of  this  section  ; 
but  for  general  purposes,  the  following  rules,  deduced  from  the 
experience  of  practical  builders  and  from  scientific  principles, 
may  be  found  useful :  these  rules  give  the  dimensions  of  the  piece 
at  its  smallest  part. 

289.  —  To  find  the  dimensions  of  a  king-post.  Rule. — Mul¬ 

tiply  the  length  of  the  post  in  feet  by  the  span  in  feet.  Then 
multiply  this  product  by  the  decimal,  0-12,  for  pine,  or  by  013 
for  oak,  which  will  give  the  area  of  the  king-post  in  inches  ;  and 
divide  this  area  by  the  breadth,  and  it  will  give  the  thickness  ;  or 
by  the  thickness  for  the  breadth.  Example. — What  should  be 
the  dimensions  of  a  pine  king-post,  8  feet  long,  fora  roof  having 
a  span  of  25  feet  ?  8  times  25  is  200 ;  this,  multiplied  by  the 

decimal,  0T2,  gives  24  inches  for  the  area;  4x6,  therefore,  would 
be  a  good  size  at  the  smallest  part. 

290.  —  To  find  the  dimensions  of  a  queen-post.  Rule. — Mul¬ 

tiply  the  length  in  feet,  of  the  queen-post  or  suspending-piece,  by 
that  part  of  the  length  of  the  tie-beam  it  supports,  also  in  feet. 
This  product,  multiplied  by  the  decimal,  0‘27,  for  pine,  or  by  0'32 
for  oak,  will  give  the  area  of  the  post  in  inches ;  and  dividing 
this  area  by  the  thickness  will  give  the  breadth.  Example. — 
The  queen-posts  in  Fig.  210  support  each  of  the  tie-beam, 
which  is  12§  feet.  To  make  them  of  pine,  6  feet  long,  what 
ghould  be  their  dimensions  ?  12f,  multiplied  by  6,  gives  76 1 


164 


AMERICAN  HOUSE-CARPENTER. 


which,  multiplied  by  0-27,  gives  2052 ;  which  indicates  a  size  of 
about  4x5*. 

291.  —  To  find  the  dimensions  of  a  tie-beam,  that  is  required 
to  support  a  ceiling  only.  Rule. — Divide  the  length  of  the 
longest  unsupported  part  by  the  cube-root  of  the  breadth  ;  and  the 
quotient,  multiplied  by  T47  for  pine,  or  by  T52  for  oak,  will  give 
the  depth  in  inches,  Example. — The  length  of  the  longest  un¬ 
supported  part  of  the  tie-beam  in  Fig.  210  is  12f  feet.  What 
should  be  the  depth  of  the  tie-beam,  the  breadth  being  6  inches, 
and  the  kind  of  wood,  pine?  The  cube-root  of  6  is  T82,  and  12f, 
divided  by  T82,  gives  a  quotient  of  6-956  ;  this,  multiplied  by 
1*47,  gives  10-225.  The  size  of  the  tie-beam,  therefore,  maybe 
6x10*.  When  there  are  rooms  in  the  roof,  the  dimensions  for 
the  tie-beam  can  be  found  by  the  rule  for  girders,  (Art.  2 77.) 

292.  —  To  find  the  dimensions  of  a  principal  rafter  when 
there  is  a  king-post  in  the  middle.  Rule. — Multiply  the  square 
of  the  length  of  the  rafter  in  feet,  by  the  span  in  feet ;  and  divide 
the  product  by  the  cube  of  the  thickness  in  inches.  For  pine, 
multiply  the  quotient  by  *096,  which  will  give  the  depth  in 
inches.  Example. — What  should  be  the  depth  of  a  rafter  of 
pine,  22-36  feet  long,  and  6  inches  thick,  the  roof  having  a  span 
of  40  feet  ?  The  square  of  22-36  is  500  nearly,  this,  multiplied  by 
40,  gives  20000  ;  and  this,  divided  by  216,  the  cube  of  the  thick¬ 
ness,  gives  92-59  ;  which,  multiplied  by  *096,  equals  S-888.  The 
size  of  the  rafter  should,  therefore,  be  6x8~. 

293.  —  To  find  the  dimensions  of  a  principal  rafter  when  two 
queen-posts  are  used  instead  of  a  king-post.  Rule. — The 
same  as  the  last,  except  that  the  decimal,  0-155,  must  be  used 
instead  of  0-96.  Example.— What  should  be  the  dimensions  of 
a  principal  rafter,  having  a  length  of  14  feet,  (as  in  Fig.  210,)  and 
a  thickness  of  6  inches,  when  the  span  of  the  roof  is  38  feet 
and  the  wood  is  pine?  The  square  of  14  is  196,  which,  multi¬ 
plied  by  38,  gives  7448 ;  this,  divided  by  216,  the  cube  of  6,  gives 


FRAMING. 


165 


34-48,  which,  multiplied  by  0-155,  gives  5-34.  The  size  of  the 
rafter  should,  therefore,  be  6x5|. 

294.  —  To  find  the  dimensions  of  a  straining-beam.  In  or¬ 
der  that  this  beam  may  be  the  strongest  possible,  its  depth  should 
be  to  its  thickness  as  10  is  to  7.  Rule. — Multiply  the  square-root 
of  the  span  in  feet,  by  the  length  of  the  straining-beam  in  feet, 
and  extract  the  square-root  of  the  product.  Multiply  this  root  by 
0-9  for  pine,  which  will  give  the  depth  in  inches  To  find  the 
thickness,  multiply  the  depth  by  the  decimal,  0-7.  Example. — 
What  should  be  the  dimensions  of  a  pine  straining-beam,  12  feet 
long,  for  a  span  of  38  feet  ?  The  square-root  of  the  span  is  6-164, 
which,  multiplied  by  12,  gives  73  968  ;  the  square-root  of  this  is 
nearly  8-60,  which,  multiplied  by  0-9,  gives  7-74 — the  depth. 
This,  multiplied  by  0-7,  gives  5-418 — the  thickness.  Therefore, 
the  beam  should  be  5 |x7|,  or  54Lx8. 

295.  —  To  find  the  dimensions  of  struts  and,  braces.  Rule. — 
Multiply  the  square-root  of  the  length  supported  in  feet,  by  the 
length  of  the  brace  or  strut  in  feet ;  and  the  square-root  of  the 
product,  multiplied  by  0-8  for  pine,  will  give  the  depth  in  inches  ; 
and  the  depth,  multiplied  by  the  decimal,  0*6,  will  give  the  thick¬ 
ness  in  inches.  Example. — In  Fig.  210,  the  part  supported  by 
the  brace  or  strut,  a,  is  equal  to  half  the  length  of  the  principal 
rafter,  or  7  feet ;  and  the  length  of  the  brace  is  6  feet :  what 
should  be  the  size  of  a  pine  brace  ?  The  square-root  of  7  is  2-65, 
which,  multiplied  by  6,  gives  15-9  ;  the  square-root  of  this  is  3-99, 
which,  multiplied  by  0-8,  gives  3-192 — the  depth.  This,  multi¬ 
plied  by  0-6,  gives  1-9152,  the  thickness.  Therefore,  the  brace 
should  be  2x3  inches. 

It  is  customary  to  make  the  principal  rafters,  lie-beam,  posts 
and  braces,  all  of  the  same  thickness,  that  the  whole  truss  may 
be  of  the  same  thickness  throughout. 

296.  —  To  find  the  dimensions  of  purlins.  Rule. — Multiply 
the  cube  of  the  length  of  the  purlin  in  feet,  by  the  distance  the 
purlins  are  apart  in  feet ;  and  the  fourth  root  of  the  product  for 
pine  will  giye  the  depth  in  inches  ;  or  multiply  by  1-04  to  obtain 


166 


AMERICAN  HOUSE-CARPENTER. 


the  depth  for  oak ;  and  the  depth,  multiplied  by  the  decimal,  06, 
will  give  the  thickness.  Example. — What  should  be  the  dimen¬ 
sions  of  pine  purlins,  9  feet  long  and  6  feet  apart  ?  The  cube  of 
9  is  729,  which,  multiplied  by  6,  gives  4374;  the  fourth  root  of 
this  is  8-13 — the  required  depth.  This,  multiplied  by  06,  gives 
4'878 — the  thickness.  A  proper  size  for  them  would  be  about 
5x8  inches.  Purlins  should  be  long  enough  to  extend  over  two, 
three  or  more  trusses. 

297.  —  To  find  the  dimensions  of  common  rafters.  The  fol¬ 

lowing  rule  is  for  slate  roofs,  having  the  rafters  placed  12  inches 
apart.  Shingle  roofs  may  have  rafters  placed  2  feet  apart.  The 
dimensions  of  rafters  for  other  kinds  of  covering  may  be  found  by 
reference  to  the  table  at  Art.  286,  and  the  laws  of  pressure  at  the 
first  part  of  this  section.  Rule. — Divide  the  length  of  bearing  in 
feet,  by  the  cube-root  of  the  breadth  in  inches  ;  and  the  quotient, 
multiplied  by  072  for  pine,  or  074  for  oak,  will  give  the  depth  in 
inches.  Example. — What  should  be  the  depth  of  a  pine  rafter, 
7  feet  long  and  2  inches  thick  ?  7  feet,  divided  by  1’26,  the  cube- 

root  of  2,  gives  5*55,  which,  multiplied  by  0.72,  gives  nearly  4 
inches — the  depth  required. 

298.  — If,  instead  of  framing  the  principal  rafters  and  straining- 
beam  into  the  king  and  the  queen  posts,  they  be  permitted  to  abut 
against  each  other,  and  the  king  and  the  queen  posts  be  made  in 
halves,  notched  on  and  bolted,  or  strapped  to  each  other  and  to  the 
tie-beam,  much  of  the  ill  effects  of  shrinking  in  the  heads  of  the 
king  and  the  queen  posts  will  be  avoided.  (See  Art.  339  and  340.) 


FRAMING. 


16? 

299. — Fig.  213  shows  a  method  of  constructing  a  truss  having 
a  built-rib  in  the  place  of  principal  rafters.  The  proper  form 
for  the  curve  is  that  of  a  parabola,  {Art.  127.)  This  curve,  when 
as  flat  as  is  described  in  the  figure,  approximates  so  near  to  that  of 
the  circle,  that  the  latter  may  be  used  in  its  stead.  The  height, 
a  b,  is  just  half  of  a  c,  the  curve  to  pass  through  the  middle  of 
the  rib.  The  rib  is  composed  of  two  series  of  abutting  pieces, 
bolted  together.  These  pieces  should  be  as  long  as  the  dimen¬ 
sions  of  the  timber  will  admit,  in  order  that  there  may  be  but  few 
joints.  The  suspending  pieces  are  in  halves,  notched  and  bolted 
to  the  tie-beam  and  rib,  and  a  purlin  is  framed  upon  the  upper  end 
of  each.  A  truss  of  this  construction  needs,  for  ordinary  roofs, 
no  diagonal  braces  between  the  suspending  pieces,  but  if  extra 
strength  is  required  the  braces  may  be  added.  The  best  place 
for  the  suspending  pieces  is  at  the  joints  of  the  rib.  A  rib  of  this 
kind  will  be  sufficiently  strong,  if  the  area  of  its  section  contain 
about  one-fourth  more  timber,  than  is  required  for  that  of  a  strain¬ 
ing-beam  for  a  roof  of  the  same  size.  The  proportion  of  the 
depth  to  the  thickness  should  be  about  as  10  is  to  7. 


300. — Some  writers  have  given  designs  for  roofs  similar  to  Fig. 
214,  having  the  tie-beam  omitted  for  the  accommodation  of  an 
arch  in  the  ceiling.  This  and  all  similar  designs  are  seriously 
objectionable,  and  should  always  be  avoided  ;  as  the  small  height 
gained  by  the  omission  of  the  tie-beam  can  never  compensate  for 
the  powerful  lateral  strains,  which  are  exerted  by  the  oblique  posi¬ 
tion  of  the  supports,  tending  to  separate  the  walls.  Where  an  arch 


168 


AMERICAN  HOUSE-CARPENTER. 


is  required  in  the  ceiling,  the  best  plan  is  to  carry  up  the  walls 
as  high  as  the  top  of  the  arch.  Then,  by  using  a  horizontal  tie- 
beam,  the  oblique  strains  will  be  entirely  removed.  Many  a  pub¬ 
lic  building  in  this  place  and  vicinity,  has  been  all  but  ruined  by 
the  settling  of  the  roof,  consequent  upon  a  defective  plan  in  the 
formation  of  the  truss  in  this  respect.  It  is  very  necessary,  there¬ 
fore,  that  the  horizontal  tie-beam  be  used,  except  where  the  walls 
are  made  so  strong  %,nd  firm  by  abutments,  or  other  support,  as  to 
prevent  a  possibility  of  their  separating. 


h 


Fig.  215. 

301. — Fig-.  215  is  a  meihod  of  obtaining  the  proper  lengths  and 
bevils  for  rafters  in  a  hip-roof,  a  b  and.  b  c  are  walls  at  the  angle 
of  the  building  ;  b  e  is  the  seat  of  the  hip-rafter  and  g  f  of  a 
jack  or  cripple  rafter.  Draw  e  h,  at  right  angles  to  b  e,  and  make 
it  equal  to  the  rise  of  the  roof;  join  b  and  h,  and  h  b  will  be  the 
length  of  the  hip-rafter.  Through  e,  draw  d  i,  at  right  angles 
to  b  c;  upon  6,  with  the  radius,  b  h ,  describe  the  arc,  h  i,  cutting 
di  ini  ;  join  b  and  i,  and  extend  gf  to  meet  b  i  in  j  ;  then  gj  will 


FRAMING. 


169 


be  the  length  of  the  jack-rafter.  The  length  of  each  jack-rafter  is 
found  in  the  same  manner — by  extending  its  seat  to  cut  the  line, 
b  i.  From  /,  draw  f  k ,  at  right  angles  to  f  g,  also  f  /,  at  right 
angles  to  be;  make  /  k  equal  to  f  l  by  the  arc,  l  k,  or  make  g  k 
equal  to  g  j  by  the  arc,  j  k  ;  then  the  angle  at  j  will  be  the  top- 
bevil  of  the  jack-rafters,  and  the  one  at  k  will  be  the  down-bevil. 

302. —  To  find  the  backing  of  the  hip-rafter.  At  any  con¬ 
venient  place  in  b  e,  (Fig.  215,)  as  o,  draw  m  n ,  at  right  angles  to 
be;  from  o,  tangical  to  b  h,  describe  a  semi-circle,  cutting  b  e  in 
5  ;  join  m  and  s  and  n  and  s ;  then  these  lines  will  form  at  s  the 
proper  angle  for  beviling  the  top  of  the  hip-rafter. 


DOMES.* 


■"•See  also  Art.  227. 
22 


iro 


AMERICAN  HOUSE-CARPENTER. 


303. — The  most  usual  form  for  domes  is  that  of  the  sphere,  the 
base  being  circular.  When  the  interior  dome  does  not  rise  too 
high,  a  horizontal  tie  may  be  thrown  across,  by  which  any  de¬ 
gree  of  strength  required  may  be  obtained.  Fig.  216  shows  a 
section,  and  Fig.  217  the  plan,  of  a  dome  of  this  kind,  a  b  being 
the  tie-beam  in  both.  Two  trusses  of  this  kind,  {Fig.  216,)  pa¬ 
rallel  to  each  other,  are  to  be  placed  one  on  each  side  of  the  open¬ 
ing  in  the  top  of  the  dome.  Upon  these  the  whole  framework  is  to 
depend  for  support,  and  their  strength  must  be  calculated  accord¬ 
ingly.  (See  the  first  part  of  this  section,  and  Art.  286.)  If  the 
dome  is  large  and  of  importance,  two  other  trusses  may  be  intro¬ 
duced  at  right  angles  to  the  foregoing,  the  tie-beams  being  pre¬ 
served  in  one  continuous  length  by  framing  them  high  enough  to 
pass  over  the  others. 


rig.  si  & 


304. — When  the  interior  dome  rises  too  high  to  admit  of  a  level 


FRAMING. 


171 


tie-beam,  the  framing  may  be  composed  of  a  succession  of  ribs 
standing  upon  a  continuous  circular  curb  of  timber,  as  seen  at 
Fig.  218  and  219, — the  latter  being  a  plan  and  the  former  a  sec¬ 
tion.  This  curb  must  be  well  secured,  as  it  serves  in  the  place 
of  a  tie-beam  to  resist  the  lateral  thrust  of  the  ribs.  In  small 
domes,  these  ribs  may  be  easily  cut  from  wide  plank  ;  but,  where 
an  extensive  structure  is  required,  they  must  be  built  in  two 
thicknesses  so  as  to  break  joints,  in  the  same  manner  as  is  descri¬ 
bed  for  a  roof  at  Art.  299.  They  should  be  placed  at  about  two 
feet  apart  at  the  base,  and  strutted  as  at  a  in  Fig.  218. 

305. — The  scantling  of  each  thickness  of  the  rib  may  be  as 
follows : 

For  domes  of  24  feet  diameter,  1x8  inches. 


it 

u 

36 

u 

lixio 

u 

a 

c 

60 

u 

2x13 

u 

a 

u 

90 

u 

2|xl3 

u 

a 

u 

108 

ii 

3x13 

a 

306. — Although  the  outer  and  the  inner  surfaces  of  a  dome 
may  be  finished  to  any  curve  that  may  be  desired,  yet  the  framing 
should  be  constructed  of  such  a  form,  as  to  insure  that  the  curve 
■of  equilibrium  will  pass  through  the  middle  of  the  depth  of  the 
framing.  The  nature  of  this  curve  is  such  that,  if  an  arch  or 
dome  be  constructed  in  accordance  with  it,  no  one  part  of  the 
structure  will  be  less  capable  than  another  of  resisting  the  strains 
and  pressures  to  which  the  whole  fabric  may  be  exposed.  The 
curve  of  equilibrium  for  an  arched  vault  or  a  roof,  where  the  load 
is  equally  diffused  over  the  whole  surface,  is  that  of  a  parabola, 
(Art.  127  ;)  for  a  dome,  having  no  lantern ,  tower  or  cupola  above 
it,  a  cubic  parabola,  (Fig.  220  ;)  and  for  one  having  a  tower,  &c., 
above  it,  a  curve  approaching  that  of  an  hyperbola  must  be  adopted, 
as  the  greatest  strength  is  required  at  its  upper  parts.  If  the 
curve  of  a  dome  be  circular,  (as  in  the  vertical  section,  Fig.  218,) 
the  pressure  will  have  a  tendency  to  burst  the  dome  outwards  at 
about  one-third  of  its  height.  Therefore,  when  this  form  is  used 


172 


AMERICAN  HOUSE-CARPENTER. 


in  the  construction  of  an  extensive  dome,  an  iron  band  should  be 
placed  around  the  framework  at  that  height ;  and  whatever  may 
be  the  form  of  the  curve,  a  band  or  tie  of  some  kind  is  necessary 
around  or  across  the  base. 

If  the  framing  be  of  a  form  less  convex  than  the  curve  of 
equilibrium,  the  weight  will  have  a  tendency  to  crush  the  ribs  in¬ 
wards,  but  this  pressure  may  be  effectually  overcome  by  strutting 
between  the  ribs  ;  and  hence  it  is  important  that  the  struts  be  so 
placed  as  to  form  continuous  horizontal  circles. 


307.  —  To  describe  a  cubic  parabola.  Let  a  b ,  {Fig,  220,)  be 
the  base  and  b  c  the  height.  Bisect  a  b  at  d,  and  divide  a  d  into 
100  equal  parts;  of  these  give  d  e  26,  ef  181,/  g  141,  g  h  121, 
h  i  lOf,  i  j  91,  and  the  balance,  8|,  to  j  a;  divide  b  c  into  8  equal 
parts,  and,  from  the  points  of  division,  draw  lines  parallel  to  a  b , 
to  meet  perpendiculars  from  the  several  points  of  division  in  a  b, 
at  the  points,  o,  o,  o,  &c.  Then  a  curve  traced  through  these 
points  will  be  the  one  required. 

308.  — Small  domes  to  light  stairways,  &c.,  are  frequently  made 
elliptical  in  both  plan  and  section ;  and  as  no  two  of  the  ribs  in 
one  quarter  of  the  dome  are  alike  in  form,  a  method  for  obtaining 
the  curves  is  necessary. 

309.  —  To  find  the  curves  for  the  ribs  of  an  elliptical  dome. 
Let  abed ,  {Fig.  221,)  be  the  plan  of  a  dome,  and  e  f  the  seat 


173 


FRAMING. 


of  one  of  the  ribs.  Then  take  e  f  for  the  transverse  axis  and 
twice  the  rise,  o  g ,  of  the  dome  for  the  conjugate,  and  describe, 
{according  to  Art.  115,  116,  &c.,)  the  semi-ellipse,  e  g f  which 
will  be  the  curve  required  for  the  rib,  e  g  f.  The  other  ribs  are 
found  in  the  same  manner. 

b  4 


310. —  To  find  the  shape  of  the  covering  for  a  spherical 
dome.  Let  A,  {Fig.  222,)  be  the  plan  and  B  the  section  of  a 
given  dome.  From  a,  draw  a  c,  at  right  angles  to  a  b  ;  find  the 
stretch-out,  {Art.  92,)  of  o  b,  and  make  d  c  equal  to  it ;  divide  the 
arc,  o  b ,  and  the  line,  d  c ,  each  into  a  like  number  of  equal  parts, 


174 


AMERICAN  HOUSE-CARPENTER. 


as  5,  (a  large  number  will  insure  greater  accuracy  than  a  small 
one ;)  upon  c,  through  the  several  points  of  division  in  c  d ,  describe 
the  arcs,  o  d  o,  1  e  1,  2/  2,  &c. ;  make  d  o  equal  to  half  the  width 
of  one  of  the  boards,  and  draw  o  s,  parallel  to  a  c  ;  join  s  and  a, 
and  from  the  points  of  division  in  the  arc,  o  b,  drop  perpendicu¬ 
lars,  meeting  a  s  in  ij  k  l ;  from  these  points,  draw  i  4,  j  3,  &c., 
parallel  to  a  c;  make  d  o,  e  l,&c.,  on  the  lower  side  of  a  c,  equal 
to  d  o,  e  1,  &c.,  on  the  upper  side ;  trace  a  curve  through  the 
points,  o,  1,  2,  3,  4,  c,  on  each  side  of  d  c  ;  then  o  c  o  will  be 
the  proper  shape  for  the  board.  By  dividing  the  circumference  of 
the  base,  A,  into  equal  parts,  and  making  the  bottom,  o  d  o,  of  the 
board  of  a  size  equal  to  one  of  those  parts,  every  board  may  be 
made  of  the  same  size.  In  the  same  manner  as  the  above,  the 
shape  of  the  covering  for  sections  of  another  form  may  be  found, 
such  as  an  ogee,  cove,  &c. 


311. —  To  find  the  curve  of  the  boards  when  laid  in  horizon¬ 
tal  courses.  Let  ABC ,  {Big.  223,)  be  the  section  of  a  given 
dome,  and  D  B  its  axis.  Divide  B  C  into  as  many  parts  as 
there  are  to  be  courses  of  boards,  in  the  points,  1,  2,  3,  &c. ; 
through  1  and  2,  draw  a  line  to  meet  the  axis  extended  at  a  ; 
then  a  will  be  the  centre  for  describing  the  edges  of  the  board,  E. 
Through  3  and  2,  draw  3  b  ;  then  b  will  be  the  centre  for  describing 
F.  Through  4  and  3,  draw  4  d ;  then  d  will  be  the  centre  for  G. 
B  is  the  centre  for  the  arc,  1  o.  If  this  method  is  taken  to  find 


FRAMING. 


175 


the  centres  for  the  boards  at  the  base  of  the  dome,  they  would 
occur  so  distant  as  to  make  it  impracticable  :  the  following  method 
is  preferable  for  this  purpose.  G  being  the  last  board  obtained  by 
the  above  method,  extend  the  curve  of  its  inner  edge  until  it 
meets  the  axis,  D  B,  in  e  ;  from  3,  through  e,  draw  3/,  meeting 
the  arc,  A  B,  inf;  join  f  and  4, /and  5  and/ and  6,  cutting  the 
axis,  D  B ,  in  s,  n  and  m  ;  from  4,  5  and  6,  draw  lines  parallel  to 
A  C  and  cutting  the  axis  in  c,  p  and  r  ;  make  c  4,  {Fig.  224,) 


equal  to  c  4  in  the  previous  figure,  and  c  s  equal  to  c  s  also  in  the 
previous  figure ;  then  describe  the  inner  edge  of  the  board,  H, 
according  to  A  rt.  87  :  the  outer  edge  can  be  obtained  by  gauging 
from  the  inner  edge.  In  like  manner  proceed  to  obtain  the  next 
board — taking  p  5  for  half  the  chord  and  p  n  for  the  height  of  the 
segment.  Should  the  segment  be  too  large  to  be  described 
easily,  reduce  it  by  finding  intermediate  points  in  the  curve,  as  at 
Art.  86. 


312. —  To  find  the  shape  of  the  angle-rib  for  a  polygonal 
dome.  Let  A  G  H,  {Fig.  225,)  be  the  plan  of  a  given  dome,  and 


176 


AMERICAN  HOUSE-CARPENTER* 


CD  a  vertical  section  taken  at  the  line,  e  f.  From  1,  2,  3,  &c., 
in  the  arc,  C  D,  draw  ordinates,  parallel  to  A  D ,  to  meet  f  G  ; 
from  the  points  of  intersection  on  /  G,  draw  ordinates  at  right- 
angles  tofG;  make  s  1  equal  to  o  1,  s  2  equal  to  o  2,  &c. ;  then 
G  f  B,  obtained  in  this  way,  will  be  the  angle-rib  required.  The 
best  position  for  the  sheathing-boards  for  a  dome  of  this  kind  is 
horizontal,  but  if  they  are  required  to  be  bent  from  the  base  to 
the  vertex,  their  shape  may  be  found  in  a  similar  manner  to  that 
shown  at  Fig-.  222. 

BRIDGES. 

313. — Various  plans  have  been  adopted  for  the  construction  of 
bridges,  of  which  perhaps  the  following  are  the  most  useful. 
Fig.  226  shows  a  method  of  constructing  wooden  bridges,  where 
the  banks  of  the  river  are  high  enough  to  permit  the  use  of  the 
tie-beam,  a  b.  The  upright  pieces,  c  d,  are  notched  and  bolted 
on  in  pairs,  for  the  support  of  the  tie-beam.  A  bridge  of  this 
construction  exerts  no  lateral  pressure  upon  the  abutments.  This 
method  may  be  employed  even  where  the  banks  of  the  river  are 
low,  by  letting  the  timbers  for  the  roadway  rest  immediately  upon 
the  tie-beam.  In  this  case,  the  framework  above  will  serve  the 
purpose  of  a  railing. 


C 


314. — Fig.  227  exhibits  a  wooden  bridge  without  a  tie-beam. 
Where  staunch  buttresses  can  be  obtained,  this  method  may  be 
recommended ;  but  if  there  is  any  doubt  of  their  stability,  it 


FRAMING. 


177 


should  not  be  attempted,  as  it  is  evident  that  such  a  system  of 
framing  is  capable  of  a  tremendous  lateral  thrust. 


315. — Fig.  228  represents  a  wooden  bridge  in  which  a  built-r  ib , 
(see  Art.  299,)  is  introduced  as  a  chief  support.  The  curve  of 
equilibrium  will  not  differ  much  from  that  of  a  parabola :  this, 
therefore^  may  be  used — especially  if  the  rib  is  made  gradually  a 
little  stronger  as  it  approaches  the  buttresses.  As  it  is  desirable 
that  a  bridge  be  kept  low,  the  following  table  is  given  to  show  the 
least  rise  that  may  be  given  to  the  rib. 


Span  in  feet. 

Least  rise  in  feet. 

Span  in  feet. 

Least  rise  in  feet. 

Span  in  feet. 

Least  rise  in  feet. 

30 

0-5 

120 

7 

280 

24 

40 

0-8 

140 

8 

300 

28 

50 

1-4 

160 

•  10 

320 

32 

60 

2 

180 

11 

350 

39 

70 

2i 

200 

12 

380 

47 

80 

3 

220 

14 

400 

53 

90 

4 

240 

17 

100 

5 

260 

20 

• 

The  rise  should  never  be  made  less  than  this,  but  in  all  cases 

23 


178 


AMERICAN  HOUSE-CARPENTER, 


greater  if  practicable ;  as  a  small  rise  requires  a  greater  quantity 
of  timber  to  make  the  bridge  equally  strong.  The  greatest  uni¬ 
form  weight  with  which  a  bridge  is  likely  to  be  loaded  is,  proba¬ 
bly,  that  of  a  dense  crowd  of  people.  This  may  be  estimated  at 
120  pounds  per  square  foot,  and  the  framing  and  gravelled  road¬ 
way  at  180  pounds  more  ;  which  amounts  to  300  pounds  on  a 
square  foot.  The  following  rule,  based  upon  this  estimate,  may 
be  useful  in  determining  the  area  of  the  ribs.  Rule. — Multiply 
the  width  of  the  bridge  by  the  square  of  half  the  span,  both  in 
feet ;  and  divide  this  product  by  the  rise  in  feet,  multiplied  by  the 
number  of  ribs ;  the  quotient,  multiplied  by  the  decimal, 
0-0011,  will  give  the  area  of  each  rib  in  feet.  When  the  road¬ 
way  is  only  planked,  use  the  decimal,  0-0007,  instead  of 
0-0011.  Example. — What  should  be  the  area  of  the  ribs  for  a 
bridge  of  200  feet  span,  to  rise  15  feet,  and  be  30  feet  wide,  with 
3  curved  ribs  ?  The  half  of  the  span  is  100  and  its  square  is 
10,000 ;  this,  multiplied  by  30,  gives  300,000,  and  15,  multi¬ 
plied  by  3,  gives  45 ;  then  300,000,  divided  by  45,  gives  6666f, 
which,  multiplied  by  0-0011,  gives  7-333  feet,  or  1056  inches  for 
the  area  of  each  rib.  Such  a  rib  may  be  24  inches  thick  by  44 
inches  deep,  and  composed  of  6  pieces,  2  in  width  and  3  in  depth. 


316. — The  above  rule  gives  the  area  of  a  rib,  that  would  be  re¬ 
quisite  to  support  the  greatest  possible  uniform  load.  But  in 
large  bridges,  a  variable  load,  such  as  a  heavy  wagon,  is  capable 
of  exerting  much  greater  strains ;  in  such  cases,  therefore,  the 
rib  should  be  made  larger.  The  greatest  concentrated  load  a 


FRAMING. 


179 


bridge  will  be  likely  to  encounter,  may  be  estimated  at  from  about 
20  to  50  thousand  pounds,  according  to  the  size  of  the  bridge. 
This  is  capable  of  exerting  the  greatest  strain,  when  placed  at 
about  one-third  of  the  span  from  one  of  the  abutments,  as  at  6, 
(Fig".  229.)  The  weakest  point  of  the  segment,  b  g  c,  is  at  g , 
the  most  distant  point  from  the  chord  line.  The  pressure  exerted 
at  b  by  the  above  weight,  may  be  considered  to  be  in  the  direction 
of  the  chord  lines,  b  a  and  be;  then,  by  constructing  the  paral¬ 
lelogram  of  forces,  e  b  f  d,  according  to  Art.  248,  b  f  will  show 
the  pressure  in  the  direction,  b  c.  Then  the  scantling  for  the  rib 
may  be  found  by  the  following  rule. 

Rule.- — Multiply  the  pressure  in  pounds  in  the  direction,  b  c, 
by  the  decimal,  0-0016,  for  white  pine,  0-0021  for  pitch  pine,  and 
0-0015  for  oak,  and  the  product  by  the  decimal  representing  the 
sine  of  the  angle,  g  b  h,  to  a  radius  of  unity.  Divide  this  pro¬ 
duct  by  the  united  breadth  in  inches  of  the  several  ribs,  and  the 
cube-root  of  the  quotient,  multiplied  by  the  distance,  b  c,  in  feet, 
will  give  the  depth  of  the  rib.  Example. — In  a  bridge  of  200 
feet  span,  15  feet  rise,  having  3  ribs  eaeh  24  inches  thick,  or  72 
inches  whole  thickness,  the  pressure  in  the  direction,  b  c,  is  found 
to  be  166,000  lbs.,  and  the  sine  of  the  angle,  g  b  h,  is  0-1 — what 
should  be  the  depth  of  the  rib  for  white  pine?  166,000,  mul¬ 
tiplied  by  0-0016,  gives  265-6,  which,  multiplied  by  0-1,  gives 
26-56  ;  this,  divided  by  72,  gives  0-3689.  The  cube-root  of  the 
last  sum  is  0-717  nearly,  and  the  distance,  b  c,  is  135  feet :  then, 
0-717,  multiplied  by  135,  gives  96f  inches,  the  depth  required. 
By  this,  each  rib  will  require  to  be  24x97  inches,  in  order  to  en¬ 
counter  without  injury  the  greatest  possible  load. 

317. — In  constructing  these  ribs,  if  the  span  be  not  over  50 
feet,  each  rib  may  be  made  in  two  or  three  thicknesses  of  timber, 
(three  thicknesses  is  preferable,)  of  convenient  lengths  bolted 
together ;  but,  in  larger  spans,  where  the  rib  will  be  such  as  to 
render  it  difficult  to  procure  timber  of  sufficient  breadth,  they 
may  be  constructed  by  bending  the  pieces  to  the  proper  curve, 


180 


AMERICAN  HOUSE-CARPENTER. 


and  bolting  them  together.  In  this  case,  where  timber  of  suffi¬ 
cient  length  to  span  the  opening  cannot  be  obtained,  and  scarfing 
is  necessary,  such  joints  must  be  made  as  will  resist  both  tension 
and  compression,  (see  Fig.  238.)  To  ascertain  the  greatest  depth 
for  the  pieces  which  compose  the  rib,  so  that  the  process  of  bend¬ 
ing  may  not  injure  their  elasticity,  multiply  the  radius  of  curvature 
in  feet  by  the  decimal,  0-05,  and  the  product  will  be  the  depth  in 
inches.  Example. — Suppose  the  curve  of  the  rib  to  be  described 
with  a  radius  of  100  feet,  then  what  should  be  the  depth  ?  The 
radius  in  feet,  100,  multiplied  by  0’05,  gives  a  product  of  5  inches. 
White  pine  or  oak  timber,  5  inches  thick,  would  freely  bend  to 
the  above  curve  ;  and,  if  the  required  depth  of  such  a  rib  be  20 
inches,  it  would  have  to  be  composed  of  at  least  4  pieces.  Pitch 
pine  is  not  quite  so  elastic  as  Avhite  pine  or  oak — its  thickness 
imay  be  found  by  using  the  decimal,  0’046,  instead  of  0’05. 


Fig.  230. 


318. — When  the  span  is  over  250  feet,  a  framed  rib,  formed  as 
in  Fig.  230,  would  be  preferable  to  the  foregoing.  Of  this,  the 
upper  and  the  lower  edges  are  formed  as  just  described,  by  bend¬ 
ing  the  timber  to  the  proper  curve.  The  pieces  that  tend  to  the 
centre  of  the  curve,  called  radials ,  are  notched  and  bolted  on  in 
pairs,  and  the  cross-braces  are  halved  together  in  the  middle,  and 
abut  end  to  end  between  the  radials.  The  distance  between  the 
ribs  of  a  bridge  should  not  exceed  about  8  feet.  The  roadway 


FRAMING. 


181 


should  be  supported  by  vertical  standards  bolted  to  the  ribs  at 
about  every  10  to  15  feet.  At  the  place  where  they  rest  on  the 
ribs,  a  double,  horizontal  tie  should  be  notched  and  bolted  on  the 
back  of  the  ribs,  and  also  another  on  the  under  side  ;  and  diago¬ 
nal  braces  should  be  framed  between  the  standards,  over  the  space 
between  the  ribs,  to  prevent  lateral  motion.  The  timbers  for  the 
roadway  may  be  as  light  as  their  situation  will  admit,  as  all  use¬ 
less  timber  is  only  an  unnecessary  load  upon  the  arch. 

319.  — It  is  found  that  if  a  roadway  be  18  feet  wide,  two  car¬ 
riages  can  pass  one  another  without  inconvenience.  Its  width, 
therefore,  should  be  either  9,  18,  27  or  36  feet,  according  to  the 
amount  of  travel.  The  width  of  the  foot-path  should  be  2  feet 
for  every  person.  When  a  stream  of  water  has  a  rapid  current, 
as  few  piers  as  practicable  should  be  allowed  to  obstruct  its 
course  ;  otherwise  the  bridge  will  be  liable  to  be  swept  away  by 
freshets.  When  the  span  is  not  over  300  feet,  and  the  banks  of 
the  river  are  of  sufficient  height  to  admit  of  it,  only  one  arch 
should  be  employed.  The  rise  of  the  arch  is  limited  by  the  form 
of  the  roadway,  and  by  the  height  of  the  banks  of  the  river; 
(See  Art.  315.)  The  rise  of  the  roadway  should  not  exceed  one 
in  24  feet,  but,  as  the  framing  settles  about  one  in  72,  the  roadway 
should  be  framed  to  rise  one  in  18,  that  it  may  be  one  in  24  after 
settling.  The  commencement  of  the  arch  at  the  abutments — the 
spring ,  as  it  is  termed,  should  not  be  below  high-water  mark  : 
and  the  bridge  should  be  placed  at  right  angles  with  the  course  of 
the  current. 

320.  — The  best  material  for  the  abutments  and  piers  of  a 
bridge,  is  stone  ;  and,  if  possible,  stone  should  be  procured  for  the 
purpose.  The  following  rule  is  to  determine  the  extent  of  the 
abutments,  they  being  rectangular,  and  built  with  stone  weighing 
120  lbs.  to  a  cubic-foot.  Rule. — Multiply  the  square  of  the 
height  of  the  abutment  by  160,  and  divide  this  product  by  the 
weight  of  a  square  foot  of  the  arch,  and  by  the  rise  of  the  arch  ; 
add  unity  to  the  quotient,  and  extract  the  square-root.  Diminish 
the  square-root  by  unity,  and  multiply  the  root,  so  diminished,  by 


182 


AMERICAN  HOUSE-CARPENTER. 


half  the  span  of  the  arch,  and  by  the  weight  of  a  square-foot  of 
the  arch.  Divide  the  last  product  by  120  times  the  height  of  the 
abutment,  and  the  quotient  will  be  the  thickness  of  the  abutment. 
Example. — Let  the  height  of  the  abutment  from  the  base  to  the 
springing  of  the  arch  be  20  feet,  half  the  span  100  feet,  the  weight 
of  a  square  foot  of  the  arch,  including  the  greatest  possible  load 
upon  it,  300  pounds,  and  the  rise  of  the  arch  18  feet — what  should 
be  its  thickness  ?  The  square  of  the  height  of  the  abutment, 
400,  multiplied  by  160,  gives  64,000,  and  300  by  18,  gives  5400  ; 
64,000,  divided  by  5400,  gives  a  quotient  of  11-852,  one  added  to 
this  makes  12-852,  the  square-root  of  which  is  3-6  ;  this,  less  one, 
is  2-6  ;  this,  multiplied  by  100,  gives  260,  and  this  again  by  300, 
gives  78,000  ;  this,  divided  by  120  times  the  height  of  the  abut¬ 
ment,  2400,  gives  32  feet  6  inches,  the  thickness  required. 

The  dimensions  of  a  pier  will  be  found  by  the  same  rule. 
For,  although  the  thrust  of  an  arch  may  be  balanced  by  an  ad¬ 
joining  arch,  when  the  bridge  is  finished,  and  while  it  remains 
uninjured ;  yet,  during  the  erection,  and  in  the  event  of  one  arch 
being  destroyed,  the  pier  should  be  capable  of  sustaining  the  en¬ 
tire  thrust  of  the  other. 

321. — Piers  are  sometimes  constructed  of  timber,  their  princi¬ 
pal  strength  depending  on  piles  driven  into  the  earth,  but  such 
piers  should  never  be  adopted  where  it  is  possible  to  avoid  them ; 
for,  being  alternately  wet  and  dry,  they  decay  much  sooner  than 
the  upper  parts  of  the  bridge.  Spruce  and  elm  are  considered 
good  for  piles.  Where  the  height  from  the  bottom  of  the 
river  to  the  roadway  is  great,  it  is  a  good  plan  to  cut  them  off  at 
a  little  below  low-water  mark,  cap  them  with  a  horizontal  tie, 
and  upon  this  erect  the  posts  for  the  support  of  the  roadway. 
This  method  cuts  off  the  part  that  is  continually  wet  from  that 
which  is  only  occasionally  so,  and  thus  affords  an  opportunity  for 
replacing  the  upper  part.  The  pieces  which  are  immersed  will 
last  a  great  length  of  time,  especially  when  of  elm ;  for  it  is  a 
well-established  fact,  that  timber  is  less  durable  when  subject  to 


a 


FRAMING. 


183 


alternate  dryness  and  moisture,  than  when  it  is  either  continually 
wet  or  continually  dry.  It  has  been  ascertained  that  the  piles 
under  London  bridge,  after  having  been  driven  about  600  years, 
were  not  materially  decayed.  These  piles  are  chiefly  of  elm,  and 
wholly  immersed. 


322. — Centres  for  stone  bridges.  Fig.  231  is  a  design  for  a 
centre  for  a  stone  bridge  where  intermediate  supports,  as  piles 
driven  into  the  bed  of  the  river,  are  practicable.  Its  timbers  are 
so  distributed  as  to  sustain  the  weight  of  the  arch-stones  as  they 
are  being  laid,  without  destroying  the  original  form  of  the  centre  ; 
and  also  to  prevent  its  destruction  or  settlement,  should  any  of  the 
piles  be  swept  away.  The  most  usual  error  in  badly-constructed 
centres  is,  that  the  timbers  are  disposed  so  as  to  cause  the  framing 
to  rise  at  the  crown,  during  the  laying  of  the  arch-stones  up  the 
sides.  To  remedy  this  evil,  some  have  loaded  the  crown  with 
heavy  stones  ;  but  a  centre  properly  constructed  will  need  no 
such  precaution. 

Experiments  have  shown  that  an  arch-stone  does  not  press 
upon  the  centring,  until  its  bed  is  inclined  to  the  horizon  at  an 
angle  of  from  30  to  45  degrees,  according  to  the  hardness  of  the 
stone,  and  whether  it  is  laid  in  mortar  or  not.  For  general  pur¬ 
poses,  the  point  at  which  the  pressure  commences,  may  be  con¬ 
sidered  to  be  at  that  joint  which  forms  an  angle  of  32  degrees 
with  the  horizon.  At  this  point,  the  pressure  is  inconsiderable, 


184 


AMERICAN  HOUSE-CARPENTER. 


but  gradually  increases  towards  the  crown.  At  an  angle  of  45 
degrees,  the  pressure  equals  about  one-quarter  the  weight  of  the 
stone  ;  at  57  degrees,  half  the  weight ;  and  when  a  vertical  line, 
as  a  b,  {Fig-.  232,)  passing  through  the  centre  of  gravity  of 


the  arch-stone,  does  not  fall  within  its  bed,  c  d,  the  pressure  may 
be  considered  equal  to  the  whole  weight  of  the  stone.  This  will 
be  the  case  at  about  60  degrees,  when  the  depth  of  the  stone  is 
double  its  breadth.  The  direction  of  these  pressures  is  consid¬ 
ered  in  a  line  with  the  radius  of  the  curve.  The  weight  upon  a 
centre  being  known,  the  pressure  may  be  estimated  and  the  tim¬ 
ber  calculated  accordingly.  But  it  must  be  remembered  that  the 
whole  weight  is  never  placed  upon  the  framing  at  once — as  seems 
to  have  been  the  idea  had  in  view  by  the  designers  of  some  cen¬ 
tres.  In  building  the  arch,  it  should  be  commenced  at  each  but¬ 
tress  at  the  same  time,  (as  is  generally  the  case,)  and  each  side 
should  progress  equally  towards  the  crown.  In  designing  the 
framing,  the  effect  produced  by  each  successive  layer  of  stone 
should  be  considered.  The  pressure  of  the  stones  upon  one  side 
should,  by  the  arrangement  of  the  struts,  be  counterpoised  by  that 
of  the  stones  upon  the  other  side. 

323. — Over  a  river  whose  stream  is  rapid,  or  where  it  is  ne¬ 
cessary  to  preserve  an  uninterrupted  passage  for  the  purposes  of 
navigation,  the  centre  must  be  constructed  without  intermediate 
supports,  and  without  a  continued  horizontal  tie  at  the  base  ;  such 
a  centre  is  shown  at  Fig.  233.  In  laying  the  stones  from  the 
base  up  to  a  and  c,  the  pieces,  b  d  and  b  d ,  act  as  ties  to  prevent 
any  rising  at  b.  After  this,  while  the  stones  are  being  laid  from 
«  and  from  c  to  b ,  they  act  as  struts  :  the  piece,  /  gy  is  added  for 


FRAMING. 


185 


b 


additional  security.  Upon  this  plan,  with  some  variation  to  suit 
circumstances,  centres  may  be  constructed  for  any  span  usual  in 
stone-bridge  building. 

324.  — In  bridge  centres,  the  principal  timbers  should  abut,  and 
not  be  intercepted  by  a  suspension  or  radial  piece  between. 
These  should  be  in  halves,  notched  on  each  side  and  bolted. 
The  timbers  should  intersect  as  little  as  possible,  for  the  more 
joints  the  greater  is  the  settling ;  and  halving  them  together  is  a 
bad  practice,  as  it  destroys  nearly  one-half  the  strength  of  the 
timber.  Ties  should  be  introduced  across,  especially  where  many 
timbers  meet ;  and  as  the  centre  is  to  serve  but  a  temporary  pur¬ 
pose,  the  whole  should  be  designed  with  a  view  to  employ  the 
timber  afterwards  for  other  uses.  For  this  reason,  all  unneces¬ 
sary  cutting  should  be  avoided. 

325.  — Centres  should  be  sufficiently  strong  to  preserve  a 
staunch  and  steady  form  during  the  whole  process  of  building ; 
for  any  shaking  or  trembling  will  have  a  tendency  to  prevent  the 
mortar  or  cement  from  setting.  For  this  purpose,  also,  the  cen¬ 
tre  should  be  lowered  a  trifle  immediately  after  the  key-stone  is 
laid,  in  order  that  the  stones  may  take  their  bearing  before  the 
mortar  is  set ;  otherwise  the  joints  will  open  on  the  Under  side. 
The  trusses,  in  centring,  are  placed  at  the  distance  of  from  4  to 
6  feet  apart,  according  to  their  strength  and  the  weight  of  the 

24 


186 


AMERICAN  HOUSE-CARPENTER. 


arch.  Between  every  two  trasses,  diagonal  braces  should  be  in¬ 
troduced  to  prevent  lateral  motion. 

326.  — In  order  that  the  centre  may  be  easily  lowered,  the  frames, 
or  trasses,  should  be  placed  upon  wedge-formed  sills  ;  as  is  shown 
at  d,  {Fig.  233.)  These  are  contrived  so  as  to  admit  of  the  settling 
of  the  frame  by  driving  the  wedge,  d,  with  a  maul,  or,  in  large 
centres,  a  piece  of  timber  mounted  as  a  battering-ram.  The 
operation  of  lowering  a  centre  should  be  very  slowly  performed, 
in  order  that  the  parts  of  the  arch  may  take  their  bearing  uni¬ 
formly.  The  wedge  pieces,  instead  of  being  placed  parallel  with 
the  truss,  are  sometimes  made  sufficiently  long  and  laid  through 
the  arch,  in  a  direction  at  right  angles  to  that  shown  at  Fig.  233. 
This  method  obviates  the  necessity  of  stationing  men  beneath  the 
arch  during  the  process  of  lowering  ;  and  was  originally  adopted 
with  success  soon  after  the  occurrence  of  an  accident,  in  lower¬ 
ing  a  centre,  by  which  nine  men  were  killed. 

327.  — To  give  some  idea  of  the  manner  of  estimating  the 
pressures,  in  order  to  select  timber  of  the  proper  scantling,  calcu¬ 
late  the  pressure  of  the  arch-stones  from  i  to  b,  {Fig.  233,)  and 
suppose  half  this  pressure  concentrated  at  a,  and  acting  in  the 
direction,  a  f.  Then,  by  reference  to  the  laws  of  pressure  and 
the  resistance  of  timber  at  Art.  248,  260,  & c.,  the  scantlings  of 
the  several  pieces  composing  the  frame,  b  d  a,  may  be  computed. 
Again,  calculate  the  pressure  of  that  portion  of  the  arch  included 
between  a  and  c,  and  consider  half  of  it  collected  at  6,  and  acting 
in  a  vertical  direction ;  then  the  amount  of  pressure  on  the  beams, 
b  d  and  b  d,  may  be  found  by  reference  to  the  first  part  of  this 
section,  as  above.  Add  the  pressure  of  that  portion  of  the  arch 
which  is  included  between  i  and  b  to  half  the  weight  of  the  cen¬ 
tre,  and  consider  this  amount  concentrated  at  tZ,  and  acting  in  a 
vertical  direction ;  then,  by  constructing  the  parallelogram  of 
forces,  the  pressure  upon  dj  may  be  ascertained. 

328.  — As  a  short  rale  for  calculating  the  scantlings  of  the  tim¬ 
bers,  let  every  strut  be  sufficiently  braced,  so  that  it  will  yield  to 


FRAMING. 


187 


crushing  before  it  will  bendunder  the  pressure — ( Art.  261.)  Then 
divide  the  pressure  in  pounds  by  1000,  and  the  quotient  will  be 
the  area  of  the  strut  in  inches.  For  example,  let  the  pressure 
upon  a  strut,  in  the  direction  of  its  axis,  be  60,000  lbs.  This, 
divided  by  1000,  gives  60,  the  area  of  the  strut  in  inches ;  the 
size  of  the  strut,  therefore,  might  be  6x10.  This  rule  is  based 
upon  experiments  by  which  it  has  been  ascertained,  that  1000 
pounds  is  the  greatest  load  that  can  be  trusted  upon  a  square  inch 
of  timber,  without  more  indentation  than  would  be  compatible 
with  the  stability  of  the  framing.  The  area  ascertained  by  the 
rule,  therefore,  must  have  reference  to  the  actual  amount  of  sur¬ 
face  upon  which  the  load  bears  ;  and  should  the  strut  have  a  tenon 
on  the  end,  the  area  of  the  shoulders,  instead  of  a  section  of  the 
whole  piece,  must  be  equal  to  the  amount  given  by  the  rule. 

329. — In  the  construction  of  arches,  the  voussoirs ,  or  arch¬ 
stones,  are  so  shaped  that  the  joints  between  them  are  perpen¬ 
dicular  to  the  curve  of  the  arch,  or  to  its  tangent  at  the  point  at 
which  the  joint  intersects  the  curve.  In  a  circular  arch,  the 
joints  tend  toward  the  centre  of  the  circle :  in  an  elliptical 
arch,  the  joints  may  be  found  by  the  following  process  : 


330.  —  To  find  the  direction  of  the  joints  for  an  elliptical 
arch.  A  joint  being  wanted  at  a,  {Fig.  234,)  draw  lines  from 
that  point  to  the  foci,  /  and /;  bisect  the  angle,  fa  f  with  the 
line,  ah  ;  then  a  h  will  be  the  direction  of  the  joint. 

331.  —  To  find  the  direction  of  the  joints  for  a  parabolic  arch. 
A  joint  being  wanted  at  a,  {Fig.  235,)  draw  a  e,  at  right  angles  to 
the  axis,  eg;  make  c  g  equal  to  c  e,  and  join  a  and  g  ;  draw  a  h,  at 
right  angles  to  a  g  ;  then  a  h  will  be  the  direction  of  the  joint. 


188 


AMERICAN  HOUSE-CARPENTER. 


The  direction  of  the  joint  from  b  is  found  in  the  same  manner. 
The  lines,  a  g  and  b  f,  are  tangents  to  the  curve  at  those  points 
respectively ;  and  any  number  of  joints  in  the  curve  may  be  ob¬ 
tained,  by  first  ascertaining  the  tangents,  and  then  drawing  lines 
at  right  angles  to  them. 


JOINTS. 


Fig.  236. 


332. — Fig.  236  shows  a  simple  and  quite  strong  method  of 
lengthening  a  tie-beam ;  but  the  strength  consists  wholly  in  the 
bolts,  and  in  the  friction  of  the  parts  produced  by  screwing  the 
pieces  firmly  together.  Should  the  timber  shrink  to  even  a  small 
degree,  the  strength  would  depend  altogether  on  the  bolts.  It 
would  be  made  much  stronger  by  indenting  the  pieces  together ; 
as  at  the  upper  edge  of  the  tie-beam  in  Fig.  237  j  or  by  placing 


Fig.  237. 


keys  in  the  joints,  as  at  the  lower  edge  in  the  same  figure.  This 
process,  however,  weakens  the  beam  in  proportion  to  the  depth 
of  the  indents. 

333. — Fig.  238  shows  a  method  of  scarfing,  or  splicing,  a  tie- 
beam  without  bolts.  The  keys  are  to  be  of  well-seasoned,  hard 


FRAMING. 


189 


<  " — ^ ?  t 

Fig.  238. 


wood,  and,  if  possible,  very  cross-grained.  The  addition  of  bolts 
would  make  this  a  very  strong  splice,  or  even  white-oak  pins 
would  add  materially  to  its  strength. 


Fig.  239. 


334.  — Fig.  239  shows  about  as  strong  a  splice,  perhaps,  as 
can  well  be  made.  It  is  to  be  recommended  for  its  simplicity ; 
as,  on  account  of  their  being  no  oblique  joints  in  it,  it  can  be 
readily  and  accurately  executed.  A  complicated  joint  is  the 
worst  that  can  be  adopted  ;  still,  some  have  proposed  joints  that 
seem  to  have  little  else  besides  complication  to  recommend 
them. 

335.  — In  proportioning  the  parts  of  these  scarfs,  the  depths  of 
all  the  indents  taken  together  should  be  equal  to  one-third  of  the 
depth  of  the  beam.  In  oak,  ash  or  elm,  the  whole  length  of  the 
scarf  should  be  six  times  the  depth,  or  thickness,  of  the  beam, 
when  there  are  no  bolts  ;  but,  if  bolts  instead  of  indents  are  used, 
then  three  times  the  breadth  ;  and,  when  both  methods  are  com¬ 
bined,  twice  the  depth  of  the  beam.  The  length  of  the  scarf  in 
pine  and  similar  soft  woods,  depending  wholly  on  indents,  should 
be  about  12  times  the  thickness,  or  depth,  of  the  beam ;  when 
depending  wholly  on  bolts,  6  times  the  breadth ;  and,  when  both 
methods  are  combined,  4  times  the  depth. 


4 


-o- 


-a- 


Fig.  240. 


336. — Sometimes  beams  have  to  be  pieced  that  are  required  to 
resist  cross  strains — such  as  a  girder,  or  the  tie-beam  of  a  roof 
when  supporting  the  ceiling.  In  such  beams,  the  fibres  of  the 


190 


AMERICAN  HOUSE-CARPENTER. 


wood  in  the  upper  part  are  compressed ;  and  therefore  a  simple  butt 
joint  at  that  place,  (as  in  Fig.  240,)  is  far  preferable  to  any  other. 
In  such  case,  an  oblique  joint  is  the  very  worst.  The  under 
side  of  the  beam  being  in  a  state  of  tension,  it  must  be  indented 
or  bolted,  or  both  ;  and  an  iron  plate  under  the  heads  of  the  bolts, 
gives  a  great  addition  of  strength. 

Scarfing  requires  accuracy  and  care,  as  all  the  indents  should 
bear  equally  ;  otherwise,  one  being  strained  more  than  another, 
there  would  be  a  tendency  to  splinter  off  the  parts.  Hence  the 
simplest  form  that  will  attain  the  object,  is  by  far  the  best.  In  all 
beams  that  are  compressed  endwise,  abutting  joints,  formed  at 
right  angles  to  the  direction  of  their  length,  are  at  once  the  simplest 
and  the  best.  For  a  temporary  purpose,  Fig.  236  would  do  very 
well ;  it  would  be  improved,  however,  by  having  a  piece  bolted 
on  all  four  sides.  Fig.  237,  and  indeed  each  of  the  others,  since 
they  have  no  oblique  joints,  would  resist  compression  well. 

337. — In  framing  one  beam  into  another  for  bearing  purposes, 
such  as  a  floor-beam  into  a  trimmer,  the  best  place  to  make  the 
mortice  in  the  trimmer,  is  in  the  neutral  line,  (see  Art.  254,) 
which  is  in  the  middle  of  its  depth.  Some  have  thought  that, 
as  the  fibres  of  the  upper  edge  are  compressed,  a  mortice  might 
be  made  there,  and  the  tenon  be  driven  in  tight  enough  to  make 
the  parts  as  capable  of  resisting  the  compression,  as  they  would 
be  without  it ;  and  they  have  therefore  concluded  that  plan  to  be 
the  best.  This  could  not  be  the  case,  even  if  the  tenon  would 
not  shrink ;  for  a  joint  between  two  pieces  cannot  possibly  be 
made  to  resist  compression,  so  well  as  a  solid  piece  without  joints. 
The  proper  place,  therefore,  for  the  mortice,  is  at  the  middle  of 
the  depth  of  the  beam ;  but  the  best  place  for  the  tenon,  in  the 
floor-beam,  is  at  its  bottom  edge.  For  the  nearer  this  is  placed  to 
the  upper  edge,  the  greater  is  the  liability  for  it  to  splinter  off;  if 
the  joint  is  formed,  therefore,  as  at  Fig.  241,  it  will  combine  all 
the  advantages  that  can  be  obtained.  Double  tenons  are  objec¬ 
tionable,  because  the  piece  framed  into  is  needlessly  weakened, 


FRAMING. 


191 


Fig.  241. 


and  the  tenons  are  seldom  so  accurately  made  as  to  bear  equally. 
For  this  reason,  unless  the  tusk  at  a  in  the  figure  fits  exactly,  so 
as  to  bear  equally  with  the  tenon,  it  had  better  be  omitted.  And 
in  sawing  the  shoulders,  care  should  be  taken  not  to  saw  into  the 
tenon  in  the  least,  as  it  would  wound  the  beam  in  the  place  least 
able  to  bear  it. 

338. — Thus  it  will  be  seen  that  framing  weakens  both  pieces, 
more  or  less.  It  should,  therefore,  be  avoided  as  much  as  possi¬ 
ble  ;  and  where  it  is  practicable  one  piece  should  rest  upon  the 
other,  rather  than  be  framed  into  it.  This  remark  applies  to  the 
bridging-joists  in  a  framed  floor,  to  the  purlins  and  jack-rafters  of 
a  roof,  &c. 


339. — In  a  framed  truss  for  a  roof,  bridge,  partition,  &c.,  the 
joints  should  be  so  constructed  as  to  direct  the  pressures  through 
the  axes  of  the  several  pieces,  and  also  to  avoid  every  tendency 
of  the  parts  to  slide.  To  attain  this  object,  the  abutting  surface 
on  the  end  of  a  strut  should  be  at  right  angles  to  the  direction  of 
the  pressure  ;  as  at  the  joint  shown  in  Fig-.  242  for  the  foot  of  a 
rafter,  (see  Art.  257,)  in  Fig.  243  for  the  head  of  a  rafter,  and  in 
Fig.  244  for  the  foot  of  a  strut  or  brace.  The  joint  at  Fig.  242 
is  not  cut  completely  across  the  tie-beam,  but  a  narrow  lip  is  left 


192 


AMERICAN  HOUSE-CARPENTER. 


standing  in  the  middle,  and.  a  corresponding  indent  is  made  in 
the  rafter,  to  prevent  the  parts  from  separating  sideways.  The 
abutting  surface  should  be  made  as  large  as  the  attainment  of 
other  necessary  objects  will  admit.  The  iron  strap  is  added  to 
prevent  the  rafter  from  sliding  out,  should  the  end  of  the  tie-beam, 
by  decay  or  otherwise,  splinter  olf.  In  making  the  joint  shown 
at  Fig.  243,  it  should  be  left  a  little  open  at  a,  so  as  to  bring  the 
parts  to  a  fair  bearing  at  the  settling  of  the  truss,  which  must 
necessarily  take  place  from  the  shrinking  of  the  king-post  and 
other  parts.  If  the  joint  is  made  fair  at  first,  when  the  truss 
settles  it  will  cause  it  to  open  at  the  under  side  of  the  rafter,  thus 
throwing  the  whole  pressure  upon  the  sharp  edge  at  a.  This  will 
cause  an  indentation  in  the  king-post,  by  which  the  truss  will  be 
made  to  settle  further ;  and  this  pressure  not  being  in  the  axis  of 
the  rafter,  it  will  be  greatly  increased,  thereby  rendering  the  rafter 
liable  to  split  and  break. 


Fig.  245.  Fig.  246.  Fig.  247. 


340. — If  the  rafters  and  struts  were  made  to  abut  end  to  end, 
as  in  Fig.  245,  246  and  247,  and  the  king  or  queen  post  notched 
on  in  halves  and  bolted,  the  ill  effects  of  shrinking  would  be 
avoided.  This  method  has  been  practised  with  success,  in  some 
of  the  most  celebrated  bridges  and  roofs  in  Europe ;  and,  were 
its  use  adopted  in  this  country,  the  unseemly  sight  of  a  hogged 
ridge  would  seldom  be  met  with.  A  plate  of  cast  iron  between 
the  abutting  surfaces,  will  equalize  the  pressure. 


FRAMING. 


193 


341.  — Fig.  248  is  a  proper  joint  for  a  collar-beam  in  a  small 
roof:  the  principle  shown  here  should  characterize  all  tie-joints. 
The  dovetail  joint,  although  extensively  practised  in  the  above 
and  similar  cases,  is  the  very  worst  that  can  be  employed.  The 
shrinking  of  the  timber,  if  only  to  a  small  degree,  permits  the  tie 
to  withdraw — as  is  shown  at  Fig.  249.  The  dotted  line  shows 
the  position  of  the  tie  after  it  has  shrunk. 

342.  — Locust  and  white-oak  pins  are  great  additions  to  the 
strength  of  a  joint.  In  many  cases,  they  would  supply  the  place 
of  iron  bolts  ;  and,  on  account  of  their  small  cost,  they  should  be 
Used  in  preference  wherever  the  strength  of  iron  is  not  requisite. 
In  small  framing,  good  cut  nails  are  of  great  service  at  the  joints  ; 
but  they  should  not  be  trusted  to  bear  any  considerable  pressure, 
as  they  are  apt  to  be  brittle.  Iron  straps  are  seldom  necessary,  as  all 
the  joinings  in  carpentry  may  be  made  without  them.  They  can 
be  used  to  advantage,  however,  at  the  foot  of  suspending-pieces, 
and  for  the  rafter  at  the  end  of  the  tie-beam.  In  roofs  for  ordi* 
nary  purposes,  the  iron  straps  for  suspending-pieces  may  be  as 
follows :  When  the  longest  unsupported  part  of  the  tie-beam  is 

10  feet,  the  strap  may  be  1  inch  wide  by  TV  thick. 

15  “  “  H  “  1  “ 

20  “  “  2  “  i  “ 

In  fastening  a  strap,  its  hold  on  the  suspending-piece  will  be  much 
increased,  by  turning  its  ends  into  the  wood.  Iron  straps  should 
be  protected  from  rust;  for  thin  plates  of  iron  decay  very  soon, 

25 


194 


AMERICAN  HOUSE-CARPENTER. 


especially  when  exposed  to  dampness.  For  this  purpose,  as  soon 
as  the  strap  is  made,  let  it  be  heated  to  about  a  blue  heat,  and, 
while  it  is  hot,  pour  over  its  entire  surface  raw  linseed  oil,  or  rub 
it  with  beeswax.  Either  of  these  will  give  it  a  coating  which 
dampness  will  not  penetrate. 


SECTION  V.— DOORS,  WINDOWS,  &c. 


DOORS. 

343. — Among  the  several  architectural  arrangements  of  an  edi¬ 
fice,  the  door  is  by  no  means  the  least  in  importance  ;  and,  if  pro¬ 
perly  constructed,  it  is  not  only  an  article  of  use,  but  also  of  or¬ 
nament,  adding  materially  to  the  regularity  and  elegance  of  the 
apartments.  The  dimensions  and  style  of  finish  of  a  door,  should 
be  in  accordance  with  the  size  and  style  of  the  building,  or  the 
apartment  for  which  it  is  designed.  As  regards  the  utility  of 
doors,  the  principal  door  to  a  public  building  should  be  of  suffi¬ 
cient  width  to  admit  of  a  free  passage  for  a  crowd  of  people ; 
while  that  of  a  private  apartment  will  be  wide  enough,  if  it  per¬ 
mit  one  person  to  pass  without  being  incommoded.  Experience 
has  determined  that  the  least  width  allowable  for  this  is  2  feet  8 
inches  ;  although  doors  leading  to  inferior  and  unimportant  rooms 
may,  if  circumstances  require  it,  be  as  narrow  as  2  feet  6  inches  ; 
and  doors  for  closets,  where  an  entrance  is  seldom  required,  may 
be  but  2  feet  wide.  The  width  of  the  principal  door  to  a  public 
building  may  be  from  6  to  12  feet,  according  to  the  size  of  the 
building  ;  and  the  width  of  doors  for  a  dwelling  may  be  from  2 
feet  8  inches,  to  3  feet  6  inches.  If  the  importance  of  an  apart¬ 
ment  in  a  dwelling  be  such  as  to  require  a  door  of  greater  width 


196  AMERICAN  HOUSE-CARPENTER. 

than  3  feet  6  inches,  the  opening  should  be  closed  with  two 
doors,  or  a  door  in  two  folds ;  generally,  in  such  cases,  where  the 
opening  is  from  5  to  8  feet,  folding  or  sliding  doors  are  adopted. 
As  to  the  height  of  a  door,  it  should  in  no  case  be  less  than  about 
6  feet  3  inches ;  and  generally  not  less  than  6  feet  8  inches. 

344.  — The  proportion  between  the  width  and  height  of  single 
doors,  for  a  dwelling,  should  be  as  2  is  to  5 ;  and,  for  entrance-* 
doors  to  public  buildings,  as  1  is  to  2.  If  the  width  is  given  and 
the  height  required  of  a  door  for  a  dwelling,  multiply  the  width 
by  5,  and  divide  the  product  by  2  ;  but,  if  the  height  is  given  and 
the  width  required,  divide  by  5,  and  multiply  by  2,  Where  two 
or  more  doors  of  different  widths  show  in  the  same  room,  it  is 
well  to  proportion  the  dimensions  of  the  more  important  by  the 
above  rule,  and  make  the  narrower  doors  of  the  same  height  as 
the  wider  ones ;  as  all  the  doors  in  a  suit  of  apartments,  except 
the  folding  or  sliding  doors,  have  the  best  appearance  when  of 
one  height.  The  proportions  for  folding  or  sliding  doors  should 
be  such  that  the  width  may  be  equal  to  |  of  the  height;  yet  this 
rule  needs  some  qualification :  for,  if  the  width  of  the  opening 
be  greater  than  one-half  the  width  of  the  room,  there  will  not  be 
a  sufficient  space  left  for  opening  the  doors ;  also,  the  height 
should  be  about  one-tenth  greater  than  that  of  the  adjacent  single 
doors. 

345.  — Where  doors  have  but  two  panels  in  width,  let  the  stiles 
and  muntins  be  each  4  °f  die  width  ;  or,  whatever  number  of 
panels  there  maybe,  let  the  united  widths  of  the  stiles  and  the 
muntins,  or  the  whole  width  of  the  solid,  be  equal  to  j  of  the  width 
of  the  door.  Thus  :  in  a  door,  35  inches  wide,  containing  two 
panels  in  width,  the  stiles  should  be  5  inches  wide  ;  and  in  a  door, 
3  feet  6  inches  wide,  the  stiles  should  be  6  inches.  If  a  door,  3 
feet  6  inches  wide,  is  to  have  3  panels  in  width,  the  stiles  and 
muntins  should  be  each  4£  inches  wide,  each  panel  being  8  inches, 
The  bottom  rail  and  the  lock  rail  ought  to  be  each  equal  in 
Width  to  TV  of  the  height  of  the  door  ;  and  the  top  rail,  and  alj 


DOORS,  WINDOWS,  <fcc. 


197 


others,  of  the  same  width  as  the  stiles.  The  moulding  on  the 
panel  should  be  equal  in  width  to  4  of  the  width  of  the  stile. 


346.  — Fig-.  250  shows  an  approved  method  of  trimming  doors  : 
a  is  the  door  stud  ;  b,  the  lath  and  plaster  ;  c,  the  ground  ;  d ,  the 
jamb  ;  e,  the  stop  ;  f  and  g,  architrave  casings  ;  and  h,  the  door 
stile.  It  is  customary  in  ordinary  work  to  form  the  stop  for  the 
door  by  rebating  the  jamb.  But,  when  the  door  is  thick  and 
heavy,  a  better  plan  is  to  nail  on  a  piece  as  at  e  in  the  figure. 
This  piece  can  be  fitted  to  the  door,  and  put  on  after  the  door  is 
hung  ;  so,  should  the  door  be  a  trifle  winding ,  this  will  correct 
the  evil,  and  the  door  be  made  to  shut  solid. 

347.  — Fig.  251  is  an  elevation  of  a  door  and  trimmings  suita¬ 
ble  for  the  best  rooms  of  a  dwelling.  (For  trimmings  generally, 
see  Sect.  III.)  The  number  of  panels  into  which  a  door  should 
be  divided,  is  adjusted  at  pleasure  ;  yet  the  present  style  of  finish¬ 
ing  requires,  that  the  number  be  as  small  as  a  proper  regard  for 
strength  will  admit.  In  some  of  our  best  dwellings,  doors  have 
been  made  having  only  two  upright  panels.  A  few  years  expe¬ 
rience,  however,  has  proved  that  the  omission  of  the  lock  rail 
is  at  the  expense  of  the  strength  and  durability  of  the  door ;  a 
four-panel  door,  therefore,  is  the  best  that  can  be  made. 

348.  — The  doors  of  a  dwelling  should  all  be  hung  so  as  to  open 
into  the  principal  rooms  ;  and,  in  general,  no  door  should  be  hung 
to  open  into  the  hall,  or  passage.  As  to  the  proper  edge  of  the 
door  on  which  to  affix  the  hinges,  no  general  rule  can  be  assigned. 


198 


AMERICAN  IIOUSE-CARPENTER. 


It  may  be  observed,  however,  that  a  bed-room  door  should  be 
hung  so  that,  when  half  open,  it  will  screen  the  bed ;  and  a  door 
leading  from  a  hall,  or  passage,  to  a  principal  room,  should  screen 
the  fire. 


WINDOWS. 

349. — A  window  should  be  of  such  dimensions,  and  in  such  a 
position,  as  to  admit  a  sufficiency  of  light  to  that  part  of  the 
apartment  for  which  it  is  designed.  No  definite  rule  for  the  size 


199 


DOORS,  WINDOWS,  &C. 

can  well  be  given,  that  will  answer  in  all  cases  ;  yet,  as  an  ap¬ 
proximation,  the  following  has  been  used  for  general  purposes. 
Multiply  together  the  length  and  the  breadth  in  feet  of  the  apart¬ 
ment  to  be  lighted,  and  the  product  by  the  height  in  feet ;  then 
the  square-root  of  this  product  will  show  the  required  number  of 
square  feet  of  glass. 

350.  — To  ascertain  the  dimensions  of  window  frames,  add 
inches  to  the  width  of  the  glass  for  their  width,  and  inches  to 
the  height  of  the  glass  for  their  height.  These  give  the  dimen¬ 
sions,  in  the  clear,  of  ordinary  frames  for  12-light  windows  ;  the 
height  being  taken  at  the  inside  edge  of  the  sill.  In  a  brick  wall, 
the  width  of  the  opening  is  8  inches  more  than  the  width  of  the 
glass — 4\  for  the  stiles  of  the  sash,  and  3^  for  hanging  stiles — 
and  the  height  between  the  stone  sill  and  lintel  is  about  10 \  inches 
more  than  the  height  of  the  glass,  it  being  varied  according  to  the 
thickness  of  the  sill  of  the  frame. 

351. - — In  hanging  inside  shutters  to  fold  into  boxes,  it  is  ne¬ 

cessary  to  have  the  box  shutter  about  one  inch  wider  than  the 
flap,  in  order  that  the  flap  may  not  interfere  when  both  are  folded 
into  the  box.  The  usual  margin  shown  between  the  face  of  the 
shutter  when  folded  into  the  box  and  the  quirk  of  the  stop  bead, 
or  edge  of  the  casing,  is  half  an  inch ;  and,  in  the  usual  method 
of  letting  the  whole  of  the  thickness  of  the  butt  hinge  into  the 
edge  of  the  box  shutter,  it  is  necessary  to  make  allowance  for  the 
throw  of  the  hinge.  This  may,  in  general,  be  estimated  at  1  of 
an  inch  at  each  hinging  ;  which  being  added  to  the  margin,  the 
entire  width  of  the  shutters  will  be  l?  inches  more  than  the  width 
of  the  frame  in  the  clear.  Then,  to  ascertain  the  width  of  the 
box  shutter,  add  1  £  inches  to  the  width  of  the  frame  in  the  clear, 
between  the  pulley  stiles  ;  divide  this  product  by  4,  and  add 
half  an  inch  to  the  quotient ;  and  the  last  product  will  be  the  re¬ 
quired  width.  For  example,  suppose  the  window  to  have  3 
lights  in  width,  11  inches  each.  Then,  3  times  11  is  33,  and  4\ 
added  for  the  wood  of  the  sash,  gives  37-f - 37  and  11  is  39, 


200 


AMERICAN  HOUSE-CARPENTER. 

and  39.  divided  by  4,  gives  9| ;  to  which  add  half  an  inch,  and 
the  result  will  be  10|  inches,  the  width  required  for  the  box  shutter. 

352. — In  disposing  and  proportioning  windows  for  the  walls  of 
a  building,  the  rules  of  architectural  taste  require  that  they  be  of 
different  heights  in  different  stories,  but  of  the  same  width.  The 
windows  of  the  upper  stories  should  all  range  perpendicularly 
over  those  of  the  first,  or  principal,  story ;  and  they  should  be 
disposed  so  as  to  exhibit  a  balance  of  parts  throughout  the  front 
of  the  building.  To  aid  in  this,  it  is  always  proper  to  place  the 
front  door  in  the  middle  of  the  front  of  the  building  ;  and,  where 
the  size  of  the  house  will  admit  of  it,  this  plan  should  be  adopted. 
(See  the  latter  part  of  Art.  214.)  The  proportion  that  the  height 
should  bear  to  the  width,  may  be,  in  accordance  with  general 
usage,  as  follows  : 

The  height  of  basement  windows,  1J  of  the  width. 


a 

u 

principal-story 

u 

2* 

a 

it 

li 

second-story 

a 

11 

a 

a 

u 

third-story 

u 

11 

a 

a 

a 

fourth-story 

a 

H 

a 

u 

li 

attic-story 

u 

the  same  as  the  width. 

But,  in  determining  the  height  of 

the  windows  for  the  several 

stories,  it  is  necessary  to  take  into  consideration  the  height  of  the 
story  in  which  the  window  is  to  be  placed.  For,  in  addition  to 
the  height  from  the  floor,  which  is  generally  required  to  be  from 
28  to  30  inches,  room  is  wanted  above  the  head  of  the  window 
for  the  window-trimming  and  the  cornice  of  the  room,  besides 
some  respectable  space  which  there  ought  to  be  between  these. 

353.— The  present  style  of  finish  requires  the  heads  of  win¬ 
dows  in  general  to  be  horizontal,  or  square-headed  ;  yet,  it  is  well 
to  be  possessed  of  information  for  trimming  circular-headed  win¬ 
dows,  as  repairs  of  these  are  occasionally  needed.  If  the  jambs 
of  a  door  or  window  be  placed  at  right  angles  to  the  face  of  the 
wall,  the  edges  of  the  soffit,  or  surface  of  the  head,  would  be 
straight,  and  its  length  be  found  by  getting  the  stretch-out  of  the 


20  L 


DOORS,  WINDOWS,  &C. 

Circle,  (Art.  92  ;)but,  when  the  jambs  are  placed  obliquely  to  the 
face  of  the  wall,  occasioned  by  the  demand  for  light  in  an 
oblique  direction,  the  form  of  the  soffit  will  be  obtained  as  in  the 
following  article  :  and,  when  the  face  of  the  wall  is  circular,  as  in 
the  succeeding  one. 


/ 


354.  —  To  find  the  form  of  the  soffit  for  circular  window- 
heads,  when  the  light  is  received  in  an  oblique  direction.  Let 
abed,  (Fig.  252,)  be  the  ground-plan  of  a  given  window,  and  e  f 
a,  a  vertical  section  taken  at  right  angles  to  the  face  of  the  jambsk 
From  a ,  through  <?,  draw  a  g,  at  right  angles  to  a  b  ;  obtain  the 
stretch-out  of  ef  a ,  and  make  e  g  equal  to  it ;  divide  e  g  and  e 
f  a ,  each  into  a  like  number  of  equal  parts,  and  drop  perpen¬ 
diculars  from  the  points  of  division  in  each  ;  from  the  points  of 
intersection,  1,  2,  3,  &c.,  in  the  line,  a  d ,  draw  horizontal  lines  to 
meet  corresponding  perpendiculars  from  eg;  then  those  points 
of  intersection  will  give  the  curve  line,  d  g,  which  will  be  the 
one  required  for  the  edge  of  the  soffit.  The  other  edge,  c  h ,  is 
found  in  the  same  manner. 

355.  —  To  find  the  form  of  the  soffit  for  circular  window- 
heads ,  when  the  face  of  the  wall  is  curved.  Let  abed ,  (Fig. 
253,)  be  the  ground-plan  of  a  given  window,  and  e  f  a,  a  vertical 
section  of  the  head  taken  at  right  angles  to  the  face  of  the  jambs. 

26 


202 


AMERICAN  HOUSE-CARPENTER. 


/ 


Proceed  as  in  the  foregoing  article  to  obtain  the  line,  d  g  ;  then 
that  will  be  the  curve  required  for  the  edge  of  the  soffit;  the 
other  edge  being  found  in  the  same  manner. 

If  the  given  vertical  section  be  taken  in  a  line  with  the  face  of 
the  wall,  instead  of  at  right  angles  to  the  face  of  the  jambs,  place 
it  upon  the  line,  c  b,  (Fig.  252 ;)  and,  having  drawn  ordinates  at 
right  angles  to  c  b,  transfer  them  to  ef  a  ;  in  this  way,  a  section 
at  right  angles  to  the  jambs  can  be  obtained. 


SECTION  VI.— STAIRS. 


356. — The  stairs  is  that  mechanical  arrangement  in  a  build¬ 
ing  by  which  access  is  obtained  from  one  story  to  another.  Their 
position,  form  and  finish,  when  determined  with  discriminating 
taste,  add  greatly  to  the  comfort  and  elegance  of  a  structure.  As 
regards  their  position,  the  first  object  should  be  to  have  them  near 
the  middle  of  the  building,  in  order  that  an  equally  easy  access 
may  be  obtained  from  all  the  rooms  and  passages.  Next  in  im¬ 
portance  is  light ;  to  obtain  which  they  would  seem  to  be  best 
situated  near  an  outer  wall,  in  which  windows  might  be  construc¬ 
ted  for  the  purpose  ;  yet  a  sky-light,  or  opening  in  the  roof,  would 
not  only  provide  light,  and  so  secure  a  central  position  for  the 
stairs,  but  may  be  made,  also,  to  assist  materially  as  an  ornament 
to  the  building,  and,  what  is  of  more  importance,  afford  an  op¬ 
portunity  for  better  ventilation. 

.  357. — It  would  seem  that  the  length  of  the  raking  side  of  the 
; pitch-board ,  or  the  distance  from  the  top  of  one  riser  to  the  top  of 
the  next,  should  be  about  the  same  in  all  cases  ;  for,  whether  stairs 
be  intended  for  large  buildings  or  for  small,  for  public  or  for  pri¬ 
vate,  the  accommodation  of  men  of  the  same  stature  is  to  be  con¬ 
sulted  in  every  instance.  But  it  is  evident  that,  with  the  same 
effort,  a  longer  step  can  be  taken  on  level  than  on  rising  ground ; 


204 


AMERICAN  HOUSE-CARPENTER. 


and  that,  although  the  tread  and  rise  cannot  be  proportioned 
merely  in  accordance  with  the  style  and  importance  of  the  build¬ 
ing,  yet  this  may  be  done  according  to  the  angle  at  which  the 
flight  rises.  If  it  is  required  to  ascend  gradually  and  easy,  the 
length  from  the  top  of  one  rise  to  that  of  another,  or  the  hypothec 
irnse  of  the  pitch-board,  may  be  long  ;  but,  if  the  flight  is  steep, 
the  length  must  be  shorter.  Upon  this  data  the  following  problem 
is  constructed. 


35S. —  To  'proportion  the  rise  and  tread  to  one  another , 
Make  the  line,  a  b ,  {Fig.  254,)  equal  to  24  inches  ;  from  b,  erect 
b  c,  at  right  angles  to  a  b ,  and  make  b  c  equal  to  12  inches  ;  join  a 
and  c,  and  the  triangle,  a  b  c ,  will  form  a  scale  upon  which  to 
graduate  the  sides  of  the  pitch-board.  For  example,  suppose  a 
very  easy  stairs  is  required,  and  the  tread  is  fixed  at  14  inches. 
Place  it  from  b  to/,  and  from/,  draw  f  g,  at  right  angles  to  a  b  ; 
then  the  length  of  / g  will  be  found  to  be  5  inches,  which  is  a 
proper  rise  for  14  inches  tread,  and  the  angle,  /  b  g,  will  show 
the  degree  of  inclination  at  which  the  flight  will  ascend.  But,  in 
a  majority  of  instances,  the  height  of  a  story  is  fixed,  while  the 
length  of  tread,  or  the  space  that  the  stairs  occupy  on  the  lower 
floor,  is  optional.  The  height  of  a  story  being  determined,  the 
height  of  each  rise  will  of  course  depend  upon  the  number  into 
which  the  whole  height  is  divided  ;  the  angle  of  ascent  being  more 
easy  if  the  number  be  great,  than  if  it  be  smaller.  By  dividing 


STAIRS. 


205 


the  whole  height  of  a  story  into  a  certain  number  of  rises,  sup¬ 
pose  the  length  of  each  is  found  to  be  6  inches.  Place  this  length 
from  b  to  h,  and  draw  h  i,  parallel  to  cib;  then  h  i,  or  b  j  will  be 
the  proper  tread  for  that  rise,  and  j  b  i  will  show  the  angle  of  as¬ 
cent.  On  the  other  hand,  if  the  angle  of  ascent  be  given,  as  a 
b  l fib  l  being  10^-  inches,  the  proper  length  of  run  for  a  step- 
ladder,)  drop  the  perpendicular,  l  k ,  from  l  to  k  ;  then  l  kb  will 
be  the  proper  proportion  for  the  sides  of  a  pitch-board  for  that 
run. 

359.  — The  angle  of  ascent  will  vary  according  to  circum¬ 
stances.  The  following  treads  will  determine  about  the  right  in¬ 
clination  for  the  different  classes  of  buildings  specified. 

In  public  edifices,  tread  about  14  inches. 

In  first-class  dwellings  “  12|  “ 

In  second-class  “  “11  “ 

In  third-class  “  and  cottages  11  9  “ 

Step-ladders  to  ascend  to  scuttles,  &c.,  should  have  from  10  to 
11  inches  run  on  the  rake  of  the  string.  (See  notes  at  Art.  103.) 

360.  — The  length  of  the  steps  is  regulated  according  to  the  ex¬ 
tent  and  importance  of  the  building  in  which  they  are  placed, 
varying  from  3  to  12  feet,  and  sometimes  longer.  Where  two  per¬ 
sons  are  expected  to  pass  each  other  conveniently,  the  shortest 
length  that  will  admit  of  it  is  3  feet ;  still,  in  crowded  cities  where 
land  is  so  valuable,  the  space  allowed  for  passages  being  very 
small,  they  are  frequently  executed  at  2^  feet. 

361.  —  To  find  the  dimensions  of  the  j)itch-board.  The  first 
thing  in  commencing  to  build  a  stairs,  is  to  make  the  pitch-hoa,t&  ; 
this  is  done  in  the  following  manner.  Obtain  very  accurately,  in 
feet  and  inches,  the  perpendicular  height  of  the  story  in  which 
the  stairs  are  to  be  placed.  This  must  be  taken  from  the  top  of 
the  floor  in  the  lower  story  to  the  top  of  the  floor  in  the  upper 
story.  Then,  to  obtain  the  number  of  rises,  the  height  in  inches 
thus  obtained  must  be  divided  by  5,  6,  7,  8,  or  9,  according  to  the 
quality  and  style  of  the  building  in  which  the  stairs  are  to  bq 


206 


AMERICAN  HOUSE-CARPENTER. 


built.  For  instance,  suppose  the  building  to  be  a  first-class 
dwelling,  and  the  height  ascertained  is  13  feet  4  inches,  or  160 
inches.  The  proper  rise  for  a  stairs  in  a  house  of  this  class  is 
about  6  inches.  Then,  160  divided  by  6,  gives  26|  inches.  This 
being  nearer  27  than  26,  the  number  of  risers,  should  be  27. 
Then  divide  the  height,  160  inches,  by  27,  and  the  quotient  will 
give  the  height  of  one  rise.  On  performing  this  operation,  the 
quotient  will  be  found  to  be  5  inches,  |  and  TV  of  an  inch. 

Then,  if  the  space  for  the  extension  of  the  stairs  is  not  limited, 
the  tread  can  be  found  as  at  Art.  358.  But,  if  the  contrary  is  the 
case,  the  whole  distance  given  for  the  treads  must  be  divided  by 
the  number  of  treads  required.  On  account  of  the  upper  floor 
forming  a  step  for  the  last  riser,  the  number  of  treads  is  always 
one  less  than  the  number  of  risers.  Having  obtained  this 
rise  and  tread,  the  pitch-board  may  be  made  in  the  follow¬ 
ing  manner.  Upon  a  piece  of  well-seasoned  board  about  §  of  an 
inch  thick,  having  one  edge  jointed  straight  and  square,  lay  the 
comer  of  a  carpenters’-square,  as  shown  at  Fig.  255.  Make  a  b 


equal  to  the  rise,  and  b  c  equal  to  the  tread ;  mark  along  those 
edges  with  a  knife,  and  cut  it  out  by  the  marks,  making  the  edges 
perfectly  square.  The  grain  of  the  wood  must  run  in  the  direction 
indicated  in  the  figure,  because,  if  it  shrinks  a  trifle,  the  rise  and 
the  tread  will  be  equally  affected  by  it.  When  a  pitch-board  is 
first  made,  the  dimensions  of  the  rise  and  tread  should  be  pre¬ 
served  in  figures,  in  order  that,  should  the  first  shrink,  a  second 
could  be  made. 

362. —  To  lay  out  the  string.  The  space  required  for  timber 


STAIRS. 


207 


c 


d 


e 


f 


a 


b 


and  plastering  under  the  steps,  is  about  5  inches  for  ordinary- 
stairs  ;  set  a  gauge,  therefore,  at  5  inches,  and  run  it  on  the  lower 
edge  of  the  plank,  as  a  b,  {Fig.  256.)  Commencing  at  one  end, 
lay  the  longest  side  of  the  pitch-board  against  the  gauge-mark,  a 
b,  as  at  c,  and  draw  by  the  edges  the  lines  for  the  first  rise  and 
tread ;  then  place  it  successively  as  at  d ,  e  and  /,  until  the  re¬ 
quired  number  of  risers  shall  be  laid  down. 


Fig.  257. 


363. — Fig.  25 7  represents  a  section  of  a  step  and  riser,  joined 
after  the  most  approved  method.  In  this,  a  represents  the  end  of 
a  block  about  2  inches  long,  two  of  which  are  glued  in  the  corner 
in  the  length  of  the  step.  The  cove  at  b  is  planed  up  square, 
glued  in,  and  stuck  after  the  glue  is  set. 


PLATFORM  STAIRS 


364. — A  platform  stairs  ascends  from  one  story  to  another  in 
two  or  more  flights,  having  platforms  between  for  resting  and 
to  change  their  direction.  This  kind  of  stairs  is  the  most  easily 
constructed,  and  is  therefore  the  most  common.  The  cylin- 


208 


AMERICAN  HOUSE-CARPENTER* 


d  Fig.  258. 


der  is  generally  of  small  diameter,  in  most  cases  about  6  inches. 
It  may  be  worked  out  of  one  solid  piece,  but  a  better  way  is  to 
glue  together  three  pieces,  as  in  Fig.  258 ;  in  which  the  pieces, 
a,  b  and  c,  compose  the  cylinder,  and  d  and  e  represent  parts  of 
the  strings.  The  strings,  after  being  glued  to  the  cylinder,  are 
secured  with  screws.  The  joining  at  o  and  o  is  the  most  proper 
for  that  kind  of  joint. 

365. —  To  obtain  the  form  of  the  lower  edge  of  the  cylinder  * 
Find  the  stretch-out,  d  e,  (Fig.  259,)  of  the  face  of  the  cylinder, 
a  b  c,  according  to  Art.  92 ;  from  d  and  e,  draw  d  f  and  e  g ,  at 
right  angles  to  d  e  ;  draw  h  g,  parallel  to  d  e,  and  make  h  f  and 
g  i,  each  equal  to  one  rise;  from  i  and/,  draw  ij  and / k,  paral¬ 
lel  to  h  g  ;  place  the  tread  of  the  pitch-board  at  these  last  lines, 
and  draw  by  the  lower  edge  the  lines,  Ic  h  and  i  l ;  parallel  to 
these,  draw  m  n  and  o  p,  at  the  requisite  distance  for  the  dimen¬ 
sions  of  the  string ;  from  s ,  the  centre  of  the  plan,  draw  5  q , 
parallel  to  df;  divide  h  q  and  q  g,  each  into  2  equal  parts,  as  at 
v  and  w  ;  from  v  and  w ,  draw  v  n  and  w  o,  parallel  to  f  d  ;  join  n 
and  o,  cutting  q  s  in  r  ;  then  the  angles,  u  n  r  and  r  o  t ,  being 
eased  off  according  to  Art.  89,  will  give  the  proper  curve  for  the 
bottom  edge  of  the  cylinder.  A  centre  may  be  found  upon  which 
to  describe  these  curves  thus  :  from  w,  draw  u  x,  at  right  angles 
to  m  n;  from  r,  draw  r  x,  at  right  angles  to  n  o  ;  then  x  will  be 
the  centre  for  the  curve,  u  r.  The  centre  for  the  curve,  r  t,  is 

found  in  the  same  manner. 

Jt 


Stairs. 


209 


366. —  To  find  the  'position  for  the  balusters.  Place  thd 
(centre  of  the  first  baluster,  (b.  Fig.  260,)  i  its  diameter  from  the 
ace  of  the  riser,  c  d ,  and  i  its  diameter  from  the  end  of  the  step, 
e  d  ;  and  place  the  centre  of  the  other  baluster,  a ,  half  the  tread 
from  the  centre  of  the  first.  The  centre  of  the  rail  must  be  placed 
over  the  centre  of  the  balusters.  Their  usual  length  is  2  feet 
5  inches,  and  2  feet  9  inches^  for  the  short  and  the  long  balusters 
respectively. 


* 


210 


AMERICAN  HOUSE-CARPENTER. 


367. —  To  find  the  face-mould  fior  a  round  hand-rail  to  plat¬ 
form  stairs.  Case  1. —  When  the  cylinder  is  small.  In  Fig. 
261,  j  and  e  represent  a  vertical  section  of  the  last  two  steps  of  the 
first  flight,  and  d  and  i  the  first  two  steps  of  the  second  flight,  of 
a  platform  stairs,  the  line,  e  /,  being  the  platform ;  and  a  b  c  is 
the  plan  of  a  line  passing  through  the  centre  of  the  rail  around 
the  cylinder.  Through  i  and  rf,  draw  i  k,  and  through  j  and  e, 
draw  j  k  ;  from  k,  draw  k  l,  parallel  to  fie;  from  b,  draw  b  m, 
parallel  to  g  a  ;  from  l,  draw  l  r,  parallel  to  k  j  ;  from  n,  draw  n 
t ,  at  right  angles  to  j  lc  ;  on  the  line,  o  6,  make  o  t  equal  to  n  t ; 
join  c  and  t :  on  the  line,  j  c,  {Fig.  262,)  make  e  c  equal  to  e  n  at 
Fig.  261 ;  from  c,  draw  c  t,  at  right  angles  to  j  c,  and  make  c  t 


STAIRS. 


211 


equal  to  c  t  at  Fig.  261  ;  through  t,  draw  p  /,  parallel  to  ;  c ,  and 
make  t  l  equal  to  1 1  at  Fig.  261  ;  join  l  and  c,  and  complete  the 
parallelogram,  eels;  find  the  points,  o,  o,  o,  according  to  Art. 
118  ;  upon  e ,  o,  o,  o,  and  l ,  successively,  with  a  radius  equal  to 
half  the  width  of  the  rail,  describe  the  circles  shown  in  the  figure  ; 
then  a  curve  traced  on  both  sides  of  these  circles  and  just  touch¬ 
ing  them,  will  give  the  proper  form  for  the  mould.  The  joint  at 
l  is  drawn  at  right  angles  to  c  l. 

368. — Elucidation  of  the  foregoing  method.  This  excellent 
plan  for  obtaining  the  face-moulds  for  the  hand-rail  of  a  platform 
stairs,  has  never  before  been  published.  It  was  communicated  to 
me  by  an  eminent  stair-builder  of  this  city :  and  having  seen 
rails  put  up  from  it,  I  am  enabled  to  give  it  my  unqualified  re¬ 
commendation.  In  order  to  have  it  fully  understood,  I  have  in¬ 
troduced  Fig.  263 ;  in  which  the  cylinder,  for  this  purpose,  is 
made  rectangular  instead  of  circular.  The  figure  gives  a  per¬ 
spective  view  of  a  part  of  the  upper  and  of  the  lower  flights,  and 
a  part  of  the  platform  about  the  cylinder.  The  heavy  lines,  i  m, 
m  c  and  cj,  show  the  direction  of  the  rail,  and  are  supposed  to 
pass  through  the  centre  of  it.  When  the  rake  of  the  second 
flight  is  the  same  as  that  of  the  first,  which  is  here  and  is  gene¬ 
rally  the  case,  the  face-mould  for  the  lower  twist  will,  when  re¬ 
versed,  do  for  the  upper  flight :  that  part  of  the  rail,  therefore, 
which  passes  from  e  to  c  and  from  c  to  l,  is  all  that  will  need  ex¬ 
planation. 

Suppose,  then,  that  the  parallelogram,  e  a  o  c,  represent  a  plane 
lying  perpendicularly  over  e  ah  f  being  inclined  in  the  direction, 
e  c,  and  level  in  the  direction,  c  o ;  suppose  this  plane,  eaoc, 


212 


AMERICAN  HOUSE-CARPENTER, 


t 


Fig.  263. 


be  revolved  on  e  c  as  an  axis,  in  the  manner  indicated  by  the  arcs, 
o  n  and  a  x ,  until  it  coincides  with  the  plane,  e  r  t  c  ;  the  line,  a 
o,  will  then  be  represented  by  the  line,  x  n  ;  then  add  the  paral¬ 
lelogram,  xrt  n,  and  the  triangle,  c  tl,  deducting  the  triangle,  ers; 
and  the  edges  of  the  plane,  e  s  l  c,  inclined  in  the  direction,  e  c,  and 
also  in  the  direction,  c  l,  will  lie  perpendicularly  over  the  plane,  e 
abf .  From  this  we  gather  that  the  line,  c  o,  being  at  right  angles  t  q 


STAIRS. 


213 


e  c,  must,  in  order  to  reach  the  point,  l,  be  lengthened  the  distance, 
n  t,  and  the  right  angle,  e  c  t,  be  made  obtuse  by  the  addition  to 
it  of  the  angle,  tel.  By  reference  to  Fig.  261,  it  will  be  seen 
that  this  lengthening  is  performed  by  forming  the  right-angled 
triangle,  cot ,  corresponding  to  the  triangle,  cot,  in  Fig.  263. 
The  line,  c  t ,  is  then  transferred  to  Fig.  262,  and  placed  at  right 
angles  to  e  c;  this  angle,  e  c  t,  being  increased  by  adding  the  an¬ 
gle,  t  cl,  corresponding  to  t  cl,  Fig.  263,  the  point,  l,  is  reached, 
and  the  proper  position  and  length  of  the  lines,  e  c  and  c  l  ob¬ 
tained.  To  obtain  the  face-mould  for  a  rail  over  a  cylindrical 
well-hole,  the  same  process  is  necessary  to  be  followed  until  the 
the  length  and  position  of  these  lines  are  found ;  then,  by  forming 
the  parallelogram,  eels,  and  describing  a  quarter  of  an  ellipse 
therein,  the  proper  form  will  be  given. 


369, — Case  2. —  When  the  cylinder  is  large.  Fig.  264  re* 


214 


AMERICAN  HOUSE-CARPENTER. 


presents  a  plan  and  a  vertical  section  of  a  line  passing  through  the 
centre  of  the  rail  as  before.  From  b,  draw  b  k,  parallel  toed ;  ex¬ 
tend  the  lines,  i  d  and  j  e,  until  they  meet  k  b  in  k  and/;  from  n, 
draw  n  l,  parallel  to  o  b  ;  through  l,  draw  l  t,  parallel  to  j  k  ;  from 
k,  draw  k  t,  at  right  angles  to  j  k  ;  on  the  line,  o  b,  make  o  t  equal 
to  k  t.  Make  e  c,  {Fig.  265.)  equal  to  e  k  at  Fig.  264  ;  from  c, 


p  s  It 


draw  c  t ,  at  right  angles  to  e  c,  and  equal  to  c  t  at  Fig.  264  ;  from 
t ,  draw  t  j?,  parallel  to  c  e,  and  make  1 1  equal  to  1 1  at  Fig.  264  ; 
complete  the  parallelogram,  eels,  and  find  the  points,  o,  o,  o,  as 
before ;  then  describe  the  circles  and  complete  the  mould  as  in 
Fig.  262.  The  difference  between  this  and  Case  1  is,  that  the 
line,  c  t,  instead  of  being  raised  and  thrown  out,  is  lowered  and 
drawn  in. 


370. — Case  3. —  Where  the  rake  meets  the  level.  In  Fig. 


STAIRS. 


215 


266,  a  b  c  is  the  plan  of  a  line  passing  through  the  centre  of  the 
rail  around  the  cylinder  as  before,  and  j  and  e  is  a  vertical  section 
of  two  steps  starting  from  the  floor,  h  g.  Bisect  e  h  in  d ,  and 
through  d,  draw  d  f  parallel  to  h  g  ;  bisect  f  n  in  l,  and  from  7, 
draw  l  7,  parallel  to  nj;  from  n,  draw  n  t,  at  right  angles  to  j  n  ; 
on  the  line,  o  b ,  make  o  t  equal  to  n  t.  Then,  to  obtain  a  mould 
for  the  twist  going  up  the  flight,  proceed  as  at  Fig.  262  ;  making 
e  c  in  that  figure  equal  to  e  n  in  Fig.  266,  and  the  other  lines  of 
a  length  and  position  such  as  is  indicated  by  the  letters  of  reference 
in  each  figure.  To  obtain  the  mould  for  the  level  rail,  extend  b 
o,  (Fig.  266,)  to  i  ;  make  o  i  equal  to  /  7,  and  join  i  and  c;  make 
c  i,  (Fig.  267,)  equal  to  c  i  at  Fig.  266  ;  through  c,  draw  c  d,  at 


o  i 


right  angles  to  c  i  ;  make  d  c  equal  to  bf  at  Fig.  266,  and  com¬ 
plete  the  parallelogram,  odd;  then  proceed  as  in  the  previous 
cases  to  find  the  mould. 

371.  — All  the  moulds  obtained  by  the  preceding  examples  have 
been  for  round  rails.  For  these,  the  mould  may  be  applied  to 
a  plank  of  the  same  thickness  as  the  rail  is  intended  to  be,  and 
the  plank  sawed  square  through,  the  joints  being  cut  square  from 
the  face  of  the  plank.  A  twist  thus  cut  and  truly  rounded  will 
hang  in  a  proper  position  over  the  plan,  and  present  a  perfect  and 
graceful  wreath. 

372.  —  To  bore  for  the  balusters  of  a  round  rail  before  round¬ 
ing  it.  Make  the  angle,  o  c  t,  (Fig.  268,)  equal  to  the  angle,  o 
c  t ,  at  Fig.  261  ;  upon  c,  describe  a  circle  with  a  radius  equal  to 
half  the  thickness  of  the  rail ;  draw  the  tangent,  b  d,  parallel  to 
t  c,  and  complete  the  rectangle,  e  b  df  having  sides  tangical  to 
the  circle ;  from  c,  draw  c  a,  at  right  angles  to  o  c ;  then,  b  d 
being  the  bottom  of  the  rail,  set  a  gauge  from  b  to  a,  and  run  it 
the  whole  length  of  the  stuff’ ;  in  boring,  place  the  centre  of  the 


216 


AMERICAN  HOUSE-CARPENTER. 


b 


bit  in  the  gauge-mark  at  a,  and  bore  in  the 'direction,  a  c.  To  do 
this  easily,  make  chucks  as  represented  in  the  figure,  the  bottom 
edge,  g  h,  being  parallel  to  o  c,  and  having  a  place  sawed  out,  as 
e  /,  to  receive  the  rail.  These  being  nailed  to  the  bench,  the  rail 
will  be  held  steadily  in  its  proper  place  for  boring  vertically. 
The  distance  apart  that  the  balusters  require  to  be,  on  the  under 
side  of  the  rail,  is  one-half  the  length  of  the  rake-side  of  the 
pitch-board. 


Fig.  269. 


STAIRS. 


217 

373. —  To  obtain ,  by  the  foregoing  principles ,  the  face-mould 
for  the  twists  of  a  moulded  rail  upon  platform  stairs.  In  Fig. 
269,  a  b  c  is  the  plan  of  a  line  passing  through  the  centre  of 
the  rail  around  the  cylinder  as  before,  and  the  lines  above 
it  are  a  vertical  section  of  steps,  risers  and  platform,  with 
the  lines  for  the  rail  obtained  as  in  Fig.  261.  Set  half  the  width 
of  the  rail  from  b  to  f  and  from  b  to  r,  and  from /  and  r,  draw/ 
e  and  r  d  parallel  to  c  a.  At  Fig.  270,  the  centre  lines  of  the 


s  d  n  e 


tail,  k  c  and  c  n,  are  obtained  as  in  the  previous  examples.  Makd 
c  i  and  cj ,  each  equal  to  c  i  at  Fig.  269,  and  draw  the  lines,  i  m 
and  y  g ,  parallel  to  c  k  ;  make  n  e  and  n  d  equal  to  n  e  and  n  d  at 
Fig.  269,  and  draw  d  o  and  e  Z,  parallel  to  n  c;  also,  through  k, 
draw  s  g,  parallel  to  n  c  ;  then,  in  the  parallelograms,  m  s  do  and 
g  s  e  Z,  find  the  elliptic  curves,  d  m  and  e  g,  according  to  Art. 
118,  and  they  will  define  the  moulds.  The  joint  is  drawn  through 
w,  at  right  angles  to  n  c,  and  is  to  be  cut  square  through  from  the 
face  of  the  plank. 


28 

0 


218 


AMERICAN  HOUSE-CARPENTER. 


374.  —  To  apply  the  mould  to  the  plank.  The  mould  obtained 
according  to  the  last  article  must  be  applied  to  both  sides  of  the 
plank,  as  shown  at  Fig.  271.  Before  applying  the  mould,  the 
edge,  e  f  must  be  bevilled  according  to  the  angle,  c  t  x,  at  Fig. 
269  ;  if  the  rail  is  to  be  canted  up,  the  edge  must  be  bevilled  at 
an  obtuse  angle  with  the  upper  face ;  but  if  it  is  to  be  canted 
down ,  the  angle  that  the  edge  makes  with  the  upper  face  must  be 
acute.  From  the  spring  of  the  curve,  a,  and  the  end,  c,  draw 
vertical  lines  across  the  edge  of  the  plank  by  applying  the  pitch- 
board,  a  b  c  ;  then,  in  applying  the  mould  to  the  other  side,  place 
the  points,  a  and  c,  at  b  and  f ;  and,  after  marking  around  it,  saw 
the  rail  out  vertically.  After  the  rail  is  sawed  out,  the  bottom 

,  and  the  top  surfaces  must  be  squared  from  the  sides. 

375.  —  To  ascertain  the  thickness  of  stuff  required  for  the 
twists.  The  thickness  of  stuff  required  for  the  twists  of  a  round 
rail,  as  before  observed,  is  the  same  as  that  for  the  straight ;  but 
for  a  moulded  rail,  the  stuff  for  the  twists  must  be  thicker  than 
that  for  the  straight.  In  Fig.  269,  draw  a  section  of  the  rail  be¬ 
tween  the  lines,  d  r  and  e  f,  and  as  close  to  the  line,  d  e ,  as  possi¬ 
ble  ;  at  the  lower  corner  of  the  section,  draw  g  h,  parallel  to  d  e  ; 
then  the  distance  that  these  lines  are  apart,  will  be  the  thickness 
required  for  the  twists  of  a  moulded  rail. 

The  foregoing  method  of  finding  moulds  for  rails  is  applicable 
to  all  stairs  which  have  continued  rails  around  cylinders,  and  are 
without  winders. 


WINDING  STAIRS. 

376.  — Winding  stairs  have  steps  tapering  narrower  at  one  end 
than  at  the  other.  In  some  stairs,  there  are  steps  of  parallel  width 
incorporated  with  tapering  steps  ;  the  former  are  then  called  flyers 
and  the  latter  winders. 

377.  —  To  describe  a  regular  geometrical  winding  stairs < 
In  Fig.  272,  abed  represents  the  inner  surface  of  the  wall  en¬ 
closing  the  space  allotted  to  the  stairs,  a  e  the  length  of  the  steps, 
and  efgh  the  cylinder,  or  face  of  the  front  string.  The  line, 


STAIRS. 


219 


a  e ,  is  given  as  the  face  of  the  first  riser,  and  the  point,  j,  for  the 
limit  of  the  last.  Make  e  i  equal  to  18  inches,  and  upon  o,  with 
o  i  for  radius,  describe  the  arc,  i  j  ;  obtain  the  number  of  risers 
and  of  treads  required  to  ascend  to  the  floor  at  j,  according  to  Art. 
361,  and  divide  the  arc,  ij,  into  the  same  number  of  equal  parts 
as  there  are  to  be  treads  ;  through  the  points  of  division,  1,  2,  3, 
&c.,  and  from  the  wall-string  to  the  front-string,  draw  lines  tend¬ 
ing  to  the  centre,  o  ;  then  these  lines  will  represent  the  face  of 
each  riser,  and  determine  the  form  and  width  of  the  steps.  Allow 
the  necessary  projection  for  the  nosing  beyond  a  e,  which  should 
be  equal  to  the  thickness  of  the  step,  and  then  a  el  k  will  be  the 
dimensions  for  each  step.  Make  a  pitch-board  for  the  wall-string 
having  a  k  for  the  tread,  and  the  rise  as  previously  ascertained  ; 
with  this,  lay  out  on  a  thicknessed  plank  the  several  risers  and 
treads,  as  at  Fig.  256,  gauging  from  the  upper  edge  of  the  string 
for  the  line  at  which  to  set  the  pitch-board. 

Upon  the  back  of  the  string,  with  al|  inch  dado  plane,  mak§ 


220 


AMERICAN  HOUSE-CARPENTER. 


a  succession  of  grooves  1|-  inches  apart,  and  parallel  with  the 
lines  for  the  risers  on  the  face.  These  grooves  must  be  cut  along 
the  whole  length  of  the  plank,  and  deep  enough  to  admit  of  the 
plank’s  bending  around  the  curve,  abed.  Then  construct  a 
drum,  or  cylinder,  of  any  common  kind  of  stuff,  and  made  to  fit 
a  curve  having  a  radius  the  thickness  of  the  string  less  than  o  a  ; 
upon  this  the  string  must  be  bent,  and  the  grooves  filled  with  strips 
of  wood,  called  keys ,  which  must  be  very  nicely  fitted  and  glued 
in.  After  it  has  dried,  a  board  thin  enough  to  bend  around  on  the 
outside  of  the  string,  must  be  glued  on  from  one  end  to  the  other, 
and  nailed  with  clout  nails.  In  doing  this,  be  careful  not  to  nail 
into  any  place  where  a  riser  or  step  is  to  enter  on  the  face. 

After  the  string  has  been  on  the  drum  a  sufficient  time  for  the 
glue  to  set,  take  it  off,  and  cut  the  mortices  for  the  steps  and 
risers  on  the  face  at  the  lines  previously  made;  which  maybe 
done  by  boring  with  a  centre-bit  half  through  the  string,  and 
nicely  chisseling  to  the  line.  The  drum  need  not  be  made  so 
large  as  the  whole  space  occupied  by  the  stairs,  but  merely  large 
enough  to  receive  one  piece  of  the  wall-string  at  once — for  it 
is  evident  that  more  than  one  will  be  required.  The  front  string 
may  be  constructed  in  the  same  manner ;  taking  e  l  instead  of  a 
k  for  the  tread  of  the  pitch-board,  dadoing  it  with  a  smaller  dado 
plane,  and  bending  it  on  a  dram  of  the  proper  size, 


a 


m 


n 


o 


V  9 

b 


378. —  To  find  the  shape  and  position  of  the  timbers  neces-. 
sary  to  support  a  winding  stairs.  The  dotted  lines  in  Fig. 
272  show  the  proper  position  of  the  timbers  as  regards  the  plan  ; 
the  shape  of  each  is  obtained  as  follows.  In  Fig.  273,  the  line, 
1  a,  is  equal  to  a  riser,  less  the  thickness  of  the  floor,  and  the 
lines,  2  m,  3  n,  4  o,  5  p  and  6  q,  are  each  equal  to  one  riser,  The 


STAIRS. 


221 


fine,  a  2,  is  equal  to  a  m  in  Fig.  272,  the  line,  m  3  to  in  n  in  that 
figure,  *&c.  In  drawing  this  figure,  commence  at  a,  and  make 
the  lines,  a  1  and  a  2,  of  the  length  above  specified,  and  draw 
them  at  right  angles  to  each  other ;  draw  2  m,  at  right  angles  to 
a  2,  and  m  3,  at  right  angles  to  m  2,  and  make  2  m  and  m  3  of 
the  lengths  as  above  specified  ;  and  so  proceed  to  the  end,  Then, 
through  the  points,  1,  2,  3,  4,  5  and  6,  trace  the  line,  lb;  upon 
the  points,  1,  2,  3,  4,  &c,,  with  the  size  of  the  timber  for  radius, 
describe  arcs  as  shown  in  the  figure,  and  by  these  the  lower  line 
may  be  traced  parallel  to  the  upper.  This  will  give  the  proper 
shape  for  the  timber,  a  b)  in  Fig.  272  ;  and  that  of  the  others  may 
be  found  in  the  same  manner,  In  ordinary  cases,  the  shape  of 
one  face  of  the  timber  will  be  sufficient,  for  a  good  workman 
can  easily  hew  it  to  its  proper  level  by  that ;  but  where  great 
accuracy  is  desirable,  a  pattern  for  the  other  side  may  be  found 
in  the  same  manner  as  for  the  first. 

379. —  To  find  the  falling-mould  for  the  rail  of  a  winding 
stairs.  In  Fig.  274,  a  cb  represents  the  plan  of  a  rail  around 
half  the  cylinder,  A  the  cap  of  the  newel,  and  1,  2,  3,  &c.,  the 
face  of  the  risers  in  the  order  they  ascend.  Find  the  stretch-out, 
e/,  of  a  c  b,  according  to  Art.  92;  from  o,  through  the  point  of 
the  mitre  at  the  newel-cap,  draw  os;  obtain  on  the  tangent,  e  d , 
the  position  of  the  points,  s  and  Id*  as  at  t  and/2 ;  from  e  t /'2  and 
f  draw  e  x,  t  if,/2  g2  and  f  h,  all  at  right  angles  to  e  d ;  make  e 
g  equal  to  one  rise  and/2  g2  equal  to  12,  as  this  line  is  drawn 
from  the  12th  riser ;  from  g)  through  g 2,  draw  g  i,  make  g  x  equal 
to  about  three-fourths  of  a  rise,  (the  top  of  the  newel,  x,  should 
be  3^-  feet  from  the  floor  ;)  draw  x  w,  at  right  angles  to  e  x ,  and 
ease  off  the  angle  at  u  ;  at  a  distance  equal  to  the  thickness  of 


*  In  the  above,  the  references,  a!2,  b-,  &c.,  are  introduced  for  the  first  time.  During  the 
time  taken  to  refer  to  the  figure,  the  memory  of  the  form  of  these  may  pass  from  the  mind, 
while  that  of  the  sound  alone  remains  ;  they  may  then  be  mistaken  for  a2,b2,  &c.  This 
pan  be  avoided  in  reading  by  giving  them  a  sound  corresponding  to  their  meaning,  which 
js  second  a  second  b,  &c.  or  a  second ,  b  second. 


.222 


AMERICAN  HOUSE-CARPENTER. 


i 


the  rail,  draw  v  w  y,  parallel  to  x  u  i;  from  the  centre  of  the  plan, 

o,  draw  o  Z,  at  right  angles  to  e  d  ;  bisect  h  n  in  p,  and  through 

p,  at  right  angles  to  g  i,  draw  a  line  for  the  joint ;  in  the  same 
manner,  draw  the  joint  at  k  ;  then  x  y  will  be  the  falling-mould 
for  that  part  of  the  rail  which  extends  from  s  to  b  on  the  plan. 

380. —  To  find  the  face-mould  for  the  rail  of  a  winding-stairs. 
From  the  extremities  of  the  joints  in  the  falling-mould,  as  k,  z 
and  y ,  {Fig.  274,)  draw  k  a2,  b 2  and  y  d,  at  right  angles  to  e  d  ; 
jnake  b  e2  equal  to  f  d.  Then,  to  obtain  the  direction  of  the 
joint,  a 2  c2,  or  b2  d\  proceed  as  at  Fig.  275,  at  which  the  parts  are 


STAIRS. 


223 


shown  at  half  their  full  size.  A  is  the  plan  of  the  rail,  and  B  is 
the  falling-mould  ;  in  which  k  z  is  the  direction  of  the  butt-joint. 
From  k ,  draw  k  6,  parallel  to  l  o,  and  k  e ,  at  right  angles  to  kb; 
from  b ,  draw  b  /,  tending  to  the  centre  of  the  plan,  and  from /,  draw 
/  e,  parallel  to  b  k  ;  from  l,  through  e,  draw  l  i,  and  from  i,  draw  i 
d ,  parallel  to  e/y  join  d  and  b ,  and  d  b  will  be  the  proper  direction 


224  AMERICAN  HOUSE-CARPENTER. 

for  the  joint  on  the  plan.  The  direction  of  the  joint  on  the  otfief 
side,  a  c,  can  be  found  by  transferring  the  distances,  x  b  and  o  d , 
to  x  a  and  o  c.  (See  Art.  384.) 


7 


Having  obtained  the  direction  of  the  joint,  make  s  r  d  b,  ( Fig v 
276,)  equal  to  s  r  d2  b 2  in  Fig.  274  ;  through  r  and  d,  draw  t  a  ; 
through  6‘  and  from  d,  draw  t  u  and  d  e ,  at  right  angles  to  i  a  ; 
make  t  u  and  d  e  equal  to  t  u  and  b 2  m,  respectively,  in  Fig.  274  ; 
from  u:  through  e,  draw  u  o  ;  through  b,  from  r,  and  from  as  many 
other  points  in  the  line,  t  a ,  as  is  thought  necessary,  as/,  h  and  /, 
draw  the  ordinates,  r  c,  f  g:  h  i,j  k  and  a  o  ;  from  u,  c,  g,  i,  k ,  e 
and  o,  draw  the  ordinates,  u  1,  c  2,  g  3,  i  4,  k  5,  e  6  and  o  7,  at 
right  angles  to  u  o  ;  make  u  1  equal  to  t  s,  c  2  equal  to  r  2,  g  3 
equal  to/  3,  &c.,  and  trace  the  curve,  1  7,  through  the  points 
thus  found  ;  find  the  curve,  c  e,  in  the  same  manner,  by  transfer¬ 
ring  the  distances  between  the  line,  t  a ,  and  the  arc,  r  d  ;  join  1 
and  c,  also  e  and  7 ;  then,  1  c  e  7  will  be  the  face-mould  required 
for  that  part  of  the  rail  which  is  denoted  by  the  letters,  s  r  d?  b 2, 
on  the  plan  at  Fig.  274. 

To  ascertain  the  mould  for  the  next  quarter,  make  acje,  {Fig. 


STAIRS* 


225 


277, )  equal  to  a*  c2  j  e 2  at  Fig.  274  ;  at  any  convenient  height  on 
the  line,  d  i,  in  that  figure,  draw  q  i2,  parallel  to  e  d ;  through  c 
andy,  {Fig.  2 77,)  draw  b  d  ;  through  a ,  and  from  y,  draw  b  k  and 
j  o,  at  right  angles  to  b  d  ;  make  b  k  andj  o  equal  to  i 2  k  and  q 
i,  respectively,  in  Fig.  274 ;  from  k,  through  o,  draw  kf;  and 
proceed  as  in  the  last  figure  to  obtain  the  face-mould,  A. 

381.  —  To  ascertain  the  requisite  thickness  of  stuff.  Case 
1. —  When  the  falling-mould  is  straight.  Make  o  h  and  k  m, 
{Fig.  277,)  equal  to  i  y  at  Fig.  274  ;  draw  h  i  and  m  n ,  parallel 
to  b  d  ;  through  the  corner  farthest  from  kf,  as  n  or  i,  draw  n  i, 
parallel  to  kf ;  then  the  distance  between  kf  and  n  i  will  give 
the  thickness  required. 

382.  — Case  2. —  When  the  falling-mould  is  curved.  In  Fig. 

278,  s  r  d  b  is  equal  to  s  r  d?  b2  in  Fig.  274.  Make  a  c  equal  to  the 
stretch-out  of  the  arc,  s  b,  according  to  Art.  92,  and  divide  a  c  and 
^  b,  each  into  a  like  number  of  equal  parts  ;  from  a  and  c,  and  from 
each  point  of  division  in  the  line,  a  c,  draw  a  k,  e  l,  &c.,  at  right  an¬ 
gles  to  a  c  ;  make  a  k  equal  to  t  u  in  Fig.  274.  and  cj  equal  to  b2  m 

29 


226 


AMERICAN  HOUSE-CARPENTER. 


in  that  figure,  and  complete  the  tailing-mould,  k  j,  every  way  equal 
to  u  m  in  Fig.  274  ;  from  the  points  of  division  in  the  arc,  s  b,  draw 
lines  radiating  towards  the  centre  of  - the  circle,  dividing  the  arc, 
r  d ,  in  the  same  proportion  as  s  b  is  divided  ;  from  d  and  b,  draw 
d  t  and  b  u ,  at  right  angles  to  a  d,  and  fromj  and  v,  draw  j  u  and  v 
id ,  at  right  angles  to  j  c  ;  then  x  t  uw  will  be  a  vertical  projection 
of  the  joint,  d  b.  Supposing  every  radiating  line  across  s  r  d  b — 
corresponding  to  the  vertical  lines  across  k  j — to  represent  a  joint, 
find  their  vertical  projection,  as  at  1,  2,  3,  4,  5  and  6  ;  through  the 
corners  of  those  parallelograms,  trace  the  curve  lines  shown  in  the 
figure  ;  then  6  u  will  he  a  helinet ,  or  vertical  projection,  of  s  r  db. 
To  find  the  thickness  of  plank  necessary  to  get  out  this  part  of 
the  rail,  draw  the  line,  z  t:  touching  the  upper  side  of  the  helinet 
in  two  places  :  through  the  corner  farthest  projecting  from  that 
line,  as  w,  draw  y  w ,  parallel  to  z  t ;  then  the  distance  between 
those  lines  will  be  the  proper  thickness  of  stuff  for  this  part  of  the 
rail.  The  same  process  is  necessary  to  find  the  thickness  of 
stuff  in  all  cases  in  which  the  falling-mould  is  in  any  way  curved. 

383. —  To  apply  the  face-mould  to  the  plank.  In  Fig.  279, 
A  represents  the  plank  with  its  best  side  and  edge  in  view,  and 
B  the  same  plank  turned  up  so  as  to  bring  in  view  the  other  side 


STAIRS. 


227 


and  the  same  edge,  this  being  square  from  the  face.  Apply  the 
tips  of  the  mould  at  the  edge  of  the  plank,  as  at  a  and  o,  (A,)  and 
mark  out  the  shape  of  the  twist ;  from  a  and  o,  draw  the  lines,  a 
b  and  o  c,  across  the  edge  of  the  plank,  the  angles,  e  a  b  and  e  o 
c,  corresponding  with  k  f  doX  Fig.  277 ;  turning  the  plank  up  as 
at  B,  apply  the  tips  of  the  mould  at  b  and  c,  and  mark  it  out  as 
shown  in  the  figure.  In  sawing  out  the  twist,  the  saw  must  be 
be  moved  in  the  direction,  a  b  ;  which  direction  will  be  perpen¬ 
dicular  when  the  twist  is  held  up  in  its  proper  position. 

In  sawing  by  the  face-mould,  the  sides  of  the  rail  are  obtained  ; 
the  top  and  bottom,  or  the  upper  and  the  lower  surfaces,  are  ob¬ 
tained  by  squaring  from  the  sides,  after  having  bent  the  falling- 
mould  around  the  outer,  or  convex  side,  and  marked  by  its  edges. 
Marking  across  by  the  ends  of  the  falling-mould  will  give  the 
position  of  the  butt-joint. 

384. — Elucidation  of  the  process  by  which  the  direction  of 
the  butt-joint  is  obtained  in  Art.  380.  Mr.  Nicholson,  in  his 
Carpenter' s  Guide ,  has  given  the  joint  a  different  direction  to 
that  here  shown  ;  he  radiates  it  towards  the  centre  of  the  cylin¬ 
der.  This  is  erroneous — as  can  be  shown  by  the  following 
operation : 

In  Fig.  280,  a  r  j  i  is  the  plan  of  a  part  of  the  rail  about  the 
joint,  s  u  is  the  stretch-out  of  a  i ,  and  gp  is  the  helinet,  or  ver¬ 
tical  projection  of  the  plan,  a  rj  i,  obtained  according  to  Art. 


228 


AMERICAN  HOUSE-CARPENTER. 


382.  Bisect  r  t,  part  of  an  ordinate  from  the  centre  of  the  plan, 
and  through  the  middle,  draw  c  b ,  at  right  angles  to  g  v  ;  from 
b  and  c,  draw  c  d  and  b  e,  at  right  angles  to  s  u  ;  from  d  and  e , 
draw  lines  radiating  towards  the  centre  of  the  plan :  then  d  o 
and  e  m  will  be  the  direction  of  the  joint  on  the  plan,  according  to 
Nicholson,  and  c  b  its  direction  on  the  falling-mould.  It  will  be 
admitted  that  all  the  lines  on  the  upper  or  the  lower  side  of  the  rail 
which  radiate  towards  the  centre  of  the  cylinder,  as  d  o,  e  m  or 
ij,  are  level ;  for  instance,  the  level  line,  w  v,  on  the  top  of  the 


STAIRS. 


229 


rail  in  the  helinet,  is  a  true  representation  of  the  radiating  line,  j  ii 
on  the  plan.  The  line,  b  h ,  therefore,  on  the  top  of  the  rail  in 
the  helinet,  is  a  true  representation  of  e  m  on  the  plan,  and  k  c  on 
the  bottom  of  the  rail  truly  represents  d  o.  From  k,  draw  k  l, 
parallel  to  c  b ,  and  from  h,  draw  h /  parallel  to  be;  join  l  and 
b,  also  c  and  /;  then  ck  lb  will  be  a  true  representation  of  the 
end  of  the  lower  piece,  B ,  and  cfhboi  the  end  of  the  upper 
piece,  A  ;  and/  k  or  h  l  will  show  how  much  the  joint  is  open  on 
the  inner,  or  concave  side  of  the  rail. 


230 


AMERICAN  HOUSE-CARPENTER, 


To  show  that  the  process  followed  in  Art.  380  is  correct,  let  d  o 
and  e  m,  {Fig.  281,)  be  the  direction  of  the  butt-joint  found  as  at 
Fig.  275.  Now,  to  project,  on  the  top  of  the  rail  in  the  helinet,  a 
line  that  does  not  radiate  towards  the  centre  of  the  cylinder,  as  j 


k,  draw  vertical  lines  from  j  and  k  to  w  and  h,  and  join  w  and  h  ; 


then  it  will  be  evident  that  w  h  is  a  true  representation  in  the  helinet 
of  j  k  on  the  plan,  it  being  in  the  same  plane  as  j  k,  and  also  in  the 
same  winding  surface  as  w  v.  The  liue,  l  n ,  also,  is  a  true  repre¬ 
sentation  on  the  bottom  of  the  helinet  of  the  line,j  Ic,  in  the  plan. 
The  line  of  the  joint,  e  m,  therefore,  is  projected  in  the  same  way 
and  truly  by  i  b  on  the  top  of  the  helinet ;  and  the  line,  d  o,  by 
c  a  on  the  bottom.  Join  a  and  i,  and  then  it  will  be  seen  that 
the  lines,  c  a,  a  i  and  i  b,  exactly  coincide  with  c  b,  the  line  of 
the  joint  on  the  convex  side  of  the  rail ;  thus  proving  the  lower 
end  of  the  upper  piece,  A,  and  the  upper  end  of  the  lower  piece, 
B,  to  be  in  one  and  the  same  plane,  and  that  the  direction  of  the 
joint  on  the  plan  is  the  true  one.  By  reference  to  Fig.  275,  it  will 
be  seen  that  the  line,  l  i,  corresponds  to  x  i  in  Fig.  281 ;  and 
that  e  k  in  that  figure  is  a  representation  of  /  b,  and  i  k  of  d  b. 


Fig.  282. 


In  getting  out  the  twists,  the  joints,  before  the  falling-mould  it 


STAIRS. 


231 


applied,  are  cut  perpendicularly,  the  face-mould  being  long  enough 
to  include  the  overplus  necessary  for  a  butt-joint.  The  face-mould 
for  A,  therefore,  would  have  to  extend  to  the  line,  i  b  ;  and  that  for 
B ,  to  the  line,  yz.  Being  sawed  vertically  at  first,  a  section  of  the 
joint  at  the  end  of  the  face-mould  for  A,  would  be  represented  in 
the  helinet  by  b  if  g.  To  obtain  the  position  of  the  line,  b  i,  on 
the  end  of  the  twist,  draw  i  s ,  {Fig.  282,)  at  right  angles  to  if ’ 
and  make  i  s  equal  to  m  e  at  Fig.  281 ;  through  s ,  draw  s  g,  pa¬ 
rallel  to  i  f  and  make  5  b  equal  to  5  b  at  Fig.  281 ;  join  b  and  *  ; 
make  i/equal  to  i  f  at  Fig.  281,  and  from  f,  draw  fg,  parallel  to  i 
b  ;  then  i  b  g  f  will  be  a  perpendicular  section  of  the  rail  over  the 
line,  e  m ,  on  the  plan  at  Fig.  281,  corresponding  to  i  b  gf  in  the 
helinet  at  that  figure  ;  and  when  the  rail  is  squared,  the  top,  or 
back,  must  be  trimmed  off  to  the  line,  i  b ,  and  the  bottom  to  the 

line,  /  g . 

385. —  To  grade  the  front  string  of  a  stairs ,  having  winders 
in  a  quarter-circle  at  the  top  of  the  flight  connected  with  flyers 
at  the  bottom.  In  Fig.  283,  a  b  represents  the  line  of  the  facia 
along  the  floor  of  the  upper  story,  bee  the  face  of  the  cylinder, 
and  c  d  the  face  of  the  front  string.  Make  g  b  equal  to  £  of  the 
diameter  of  the  baluster,  and  draw  the  centre-line  of  the  rail,  fg, 
g  h  i  and  ij,  parallel  to  a  b,  b  e  c  and  c  d ;  make  g  k  and  g  l 
each  equal  to  half  the  width  of  the  rail,  and  through  k  and  l, 
draw  lines  for  the  convex  and  the  concave  sides  of  the  rail,  parallel 
to  the  centre-line ;  tangical  to  the  convex  side  of  the  rail,  and  parallel 
to  k  m ,  draw  no;  obtain  the  stretch-out,  g  r,  of  the  semi-circle,  k 
p  m,  according  to  Art.  92 ;  extend  a  b  to  t,  and  k  m  to  s  ;  make  c  5 
equal  to  the  length  of  the  steps,  and  i  u  equal  to  18  inches,  and  de¬ 
scribe  the  arcs,  s  t  and  u  6,  parallel  to  mp  ;  from  t ,  draw  t  w ,  tend¬ 
ing  to  the  centre  of  the  cylinder  ;  from  6,  and  on  the  line,  6  ux ,  run 
off  the  regular  tread,  as  at  5,  4,  3,  2,  1  and  v  ;  make  u  x  equal  to 
half  the  arc,  u  6,  and  make  the  point  of  division  nearest  to  x,  as 
v,  the  limit  of  the  parallel  steps,  or  flyers  ;  make  r  o  equal  to  mz  ; 
from  o,  draw  o  a2.  at  right  angles  to  n  o.  and  equal  to  one  rise ; 


232 


AMERICAN  HOUSE-CARPENTER. 


n3  hs 


from  a2,  draw  a2  s,  parallel  to  n  o ,  and  equal  to  one  tread ;  from  s7 
through  o,  draw  s  b 2. 

Then  from  w,  draw  w  c2,  at  right  angles  to  n  o,  and  set  up,  on 
the  line,  w  c2,  the  same  number  of  risers  that  the  floor,  A,  is  above 
the  first  winder,  B ,  as  at  1,  2,  3,  4,  5  and  6 ;  through  5,  (on  the 
arc,  6  w,)  draw  d2  e2,  tending  to  the  centre  of  the  cylinder ;  from 
e2,  draw  e2/2,  at  right  angles  to  n  o,  and  through  5,  (on  the  line, 


STAIRS. 


233 


w  c2,)  draw  g2f2,  parallel  to  no  ;  through  6,  (on  the  line,  w  c2,) 
and /2,  draw  the  line,  A2  b2 ;  make  6  c2  equal  to  half  a  rise,  and 
from  c2  and  6,  draw  c2  i2  and  6/,  parallel  to  n  o  ;  make  A2  i2  equal 
to  A2/2 ;  from  i2,  draw  i2  k2,  at  right  angles  to  i2  A 2,  and  from/2, 
draw  /2  k2,  at  right  angles  to  f1  A 2 ;  upon  k\  with  k2  f2  for  radius, 
describe  the  arc,/2  i2;  make  b2  P  equal  to  b2/2,  and  ease  off  the 
angle  at  b2  by  the  curve,  / 2  P.  In  the  figure,  the  curve  is  de¬ 
scribed  from  a  centre,  but  in  a  full-size  plan,  this  would  be  imprac¬ 
ticable  ;  the  best  way  to  ease  the  angle,  therefore,  would  be  with 
a  tanged  curve,  according  to  Art.  89.  Then  from  1,  2,  3  and  4, 
(on  the  line,  w  c2,)  draw  lines  parallel  to  n  o,  meeting  the  curve  in 
m2,  ?t2,  o2  and  p 2 ;  from  these  points,  draw  lines  at  right  angles  to 
n  o,  and  meeting  it  in  x2,  r2,  s2  and  t2 ;  from  x2  and  r2,  draw  lines 
tending  to  u2,  and  meeting  the  convex  side  of  the  rail  in  y2  and 
z2 ;  make  m  v2  equal  to  r  s2,  and  m  w2  equal  to  r  t2 ;  from  y2,  z2, 
v2,  and  w2,  through  4,  3,  2  and  1,  draw  lines  meeting  the  line  of 
the  wall-string  in  a3,  b 3,  c3  and  d3 ;  from  e3,  where  the  centre-line  of 
the  rail  crosses  the  line  of  the  floor,  draw  e3/3,  at  right  angles  to  n 
o,  and  from/3,  through  6,  draw /3  g2 ;  then  the  heavy  lines,/3 g\ 
e2  cP,  y2  a 3,  z2  b 3,  v 2  c3,  w 2  cP ,  and  z  y ,  will  be  the  lines  for  the  risers, 
which,  being  extended  to  the  line  of  the  front  string,  b  ec  d,  will 
give  the  dimensions  of  the  winders,  and  the  grading  of  the  front 
string,  as  was  required. 

386. —  To  obtain  the  falling-mould  for  the  heists  of  the  last- 
mentioned  stairs.  Make  i2  g3  and  i2  A3,  ( Fig.  283,)  each  equal 
to  half  the  thickness  of  the  rail ;  through  h3  and  g3,  draw  h3  i3 
and  g3f,  parallel  to  i2  s  ;  assuming  k  k3  and  m  m3  on  the  plan  as 
the  amount  of  straight  to  be  got  out  with  the  twists,  make  n  q 
equal  to  k  k3,  and  r  l3  equal  to  mm3;  from  n  and  P,  draw  lines  at 
right  angles  to  n  o,  meeting  the  top  of  the  falling-mould  in  n3  and 
o3 ;  from  o3,  draw  a  line  crossing  the  falling-mould  at  right  angles 
to  a  chord  of  the  curve,/2  P ;  through  the  centre  of  the  cylinder, 
draw  u2  8,  at  right  angles  to  n  o  ;  through  8,  draw  7  9,  tending  to 
IP  ;  then  n 3  7  will  be  the  falling-mould  for  the  upper  twist,  and  7 
o3  the  falling-mould  for  the  lower  twist. 

30 


234 


AMERICAN  HOUSE-CARPENTER. 


387. —  To  obtain  the  face-moulds.  The  moulds  for  the  twists 
of  this  stairs  may  be  obtained  as  at  Art.  380  ;  but,  as  the  falling- 
mould  in  its  course  departs  considerably  from  a  straight  line,  it 
would,  according  to  that  method,  require  a  very  thick  plank  for 
the  rail,  and  consequently  cause  a  great  waste  of  stuff.  In  order, 
therefore,  to  economize  the  material,  the  following  method  is  to 
be  preferred — in  which  it  will  be  seen  that  the  heights  are  taken 
in  three  places  instead  of  two  only,  as  is  done  in  the  previous 
method. 


Case  1. —  When  the  middle  height  is  above  a  line  joining 
the  other  two.  Having  found  at  Fig.  283  the  direction  of  the 
joint,  w  s 3  and  p  e,  according  to  Art.  380,  make  k  p  e  a,  {Fig. 
284,)  equal  to  k3  p3  e  p  in  Fig.  283  ;  join  b  and  c,  and  from  o , 
draw  o  h ,  at  right  angles  to  b  c  ;  obtain  the  stretch-out  of  d  g,  as 
df  and  at  Fig.  283,  place  it  from  the  axis  of  the  cylinder,  p ,  to 
q3  ;  from  q 3  in  that  figure,  draw  q3  r3,  at  right  angles  to  n  o  ;  also, 
at  a  convenient  height  on  the  line,  n  n 3,  in  that  figure,  and  at 
right  angles  to  that  line,  draw  u 3  v3 ;  from  b  and  c,  in  Fig.  284, 


STAIRS. 


235 


draw  b  j  and  c  Z,  at  right  angles  to  be;  make  b  j  equal  to  u?  n3  in 
Fig.  283,  i  h  equal  to  w3  r3  in  that  figure,  and  c  l  equal  to  v3  9  ; 
from  Z,  through  j,  draw  l  m  ;  from  h,  draw  h  n,  parallel  to  c  b  ; 
from  n,  draw  n  r ,  at  right  angles  to  b  c,  and  join  r  and  s  ;  through 
the  lowest  corner  of  the  plan,  as  p,  draw  v  e,  parallel  to  be;  from 
a,  e,  u ,  p,  k ,  t ,  and  from  as  many  other  points  as  is  thought  ne¬ 
cessary,  draw  ordinates  to  the  base-line,  v  e,  parallel  to  r  s ; 
through  h,  draw  w  x,  at  right  angles  to  ml;  upon  n,  with  r  s  for 
radius,  describe  an  intersecting  arc  at  x,  and  join  n  and  x  ;  from 
the  points  at  which  the  ordinates  from  the  plan  meet  the  base¬ 
line.,  v  e,  draw  ordinates  to  meet  the  line,  m  Z,  at  right  angles  to  v 
e  ;  and  from  the  points  of  intersection  on  m  Z,  draw  correspond¬ 
ing  ordinates,  parallel  to  n  x  ;  make  the  ordinates  which  are  pa¬ 
rallel  to  n  x  of  a  length  corresponding  to  those  which  are  parallel 
to  r  5,  and  through  the  points  thus  found,  trace  the  face-mould 
as  required. 

Case  2. —  When  the  middle  height  is  below  a  line  joining 
the  other  two.  The  lower  twist  in  Fig.  283  is  of  this  nature. 
The  face-mould  for  this  is  found  at  Fig.  285  in  a  manner  similar 
to  that  at  Fig.  284.  The  heights  are  all  taken  from  the  top  of 
the  falling-mould  at  Fig.  283 ;  b  j  being  equal  to  w  6  in  Fig.  283, 
i  h  equal  to  x3  y3  in  that  figure,  and  c  l  to  Z3  o3.  Draw  a  line 
through  j  and  Z,  and  from  h,  draw  h  n,  parallel  to  be;  from  n, 
draw  n  r,  at  right  angles  to  b  c,  and  join  r  and  s  ;  then  r  s  will  be 
the  bevil  for  the  lower  ordinates.  From  /t,  draw  h  x,  at  right  an¬ 
gles  to  j  l  ;  upon  n,  with  r  s  for  radius,  describe  an  intersecting 
arc  at  x ,  and  join  n  and  x  ;  then  n  x  will  be  the  bevil  for  the  upper 
ordinates,  upon  which  the  face-mould  is  found  as  in  Case  1. 

388. — Elucidation  of  the  foregoing  method. — This  method 
of  finding  the  face-moulds  for  the  handrailing  of  winding  stairs, 
being  founded  on  principles  which  govern  cylindric  sections,  may 
be  illustrated  by  the  following  figures.  Fig.  286  and  287  repre¬ 
sent  solid  blocks,  or  prisms,  standing  upright  on  a  level  base,  b  d  ; 
the  upper  surface,  j  a  forming  oblique  angles  with  the  face,  b  l — 


236 


AMERICAN  HOUSE-CARPENTER. 


in  Fig.  286  obtuse,  and  in  Fig.  287  acute.  Upon  the  base,  de¬ 
scribe  the  semi-circle,  b  s  c  ;  from  the  centre,  i ,  draw  i  s ,  at  right 
angles  to  6  c  ;  from  s,  draw  s  x,  at  right  angles  to  e  d,  and  from  i, 
draw  i  h,  at  right  angles  to  be;  make  i  h  equal  to  s  x,  and  join 
h  and  x  ;  then,  h  and  x  being  of  the  same  height,  the  line,  h  x , 
joining  them,  is  a  level  line.  From  h ,  draw  h  n ,  parallel  to  b  c, 
and  from  n,  draw  n  r,  at  right  angles  to  be;  join  r  and  s ,  also  n 


STAIRS. 


237 


and  x  ;  then,  n  and  x  being  of  the  same  height,  n  x  is  a  level  line ; 
and  this  line  lying  perpendicularly  over  r  s,  n  x  and  r  s  must  be 
of  the  same  length.  So,  all  lines  on  the  top,  drawn  parallel  to  n 
x,  and  perpendicularly  over  corresponding  lines  drawn  parallel  to 
r  s  on  the  base,  must  be  equal  to  those  lines  on  the  base  ;  and  by 
drawing  a  number  of  these  on  the  semi-circle  at  the  base  and 
others  of  the  same  length  at  the  top,  it  is  evident  that  a  curve,  j 
x  l,  may  be  traced  through  the  ends  of  those  on  the  top,  which 
shall  lie  perpendicularly  over  the  semi-circle  at  the  base. 

It  is  upon  this  principle  that  the  process  at  Fig.  284  and  285 
is  founded.  The  plan  of  the  rail  at  the  bottom  of  those  figures 
is  supposed  to  lie  perpendicularly  under  the  face-mould  at  the  top ; 
and  each  ordinate  at  the  top  over  a  corresponding  one  at  the  base. 
The  ordinates,  n  x  and  r  s,  in  those  figures,  correspond  to  n  x 
and  r  s  in  Fig.  286  and  287. 

In  Fig.  288,  the  top,  e  a,  forms  a  right  angle  with  the  face,  d 
c  ;  all  that  is  necessary,  therefore,  in  this  figure,  is  to  find  a  line 
corresponding  to  h  x  in  the  last  two  figures,  and  that  will  lie  level 
and  in  the  upper  surface ;  so  that  all  ordinates  at  right  angles  to 
dr  on  the  base,  will  correspond  to  those  that  are  at  right  angles 


238 


AMERICAN  HOUSE-CARPENTER. 


to  e  c  on  the  top.  This  elucidates  Fig.  276  ;  at  which  the  lines, 
h  9  and  i  8,  correspond  to  A  9  and  i  8  in  this  figure. 


389. —  To  find  the  bevil  for  the  edge  of  the  plank.  The 
plank,  before  the  face-mould  is  applied,  must  be  bevilled  accord¬ 
ing  to  the  angle  which  the  top  of  the  imaginary  block,  or  prism, 
in  the  previous  figures,  makes  with  the  face.  This  angle  is  de¬ 
termined  in  the  following  manner  :  draw  w  i,  {Fig.  289,)  at  right 
angles  to  i  s,  and  equal  to  w  h  at  Fig.  284 ;  make  i  s  equal  to  i  s  in 
that  figure,  and  join  w  and  s  ;  then  sw  p  will  be  the  bevil  required 
in  order  to  apply  the  face-mould  at  Fig.  284.  In  Fig.  285,  the 
middle  height  being  below  the  line  joining  the  other  two,  the  bevil 
is  therefore  acute.  To  determine  this,  draw  i  s,  {Fig.  290,)  at 


STAIRS. 


239 


right  angles  to  i  p,  and  equal  to  i  s  in  Fig \  285  ;  make  s  10  equal 
to  h  w  in  Fig.  285,  and  join  w  and  i  ;  then  w  i  p  will  be  the 
bevil  required  in  order  to  apply  the  face-mould  at  Fig.  285.  Al¬ 
though  the  falling-mould  in  these  cases  is  curved,  yet,  as  the 
plank  is  sprung ,  or  bevilled  on  its  edge,  the  thickness  necessary 
to  get  out  the  twist  may  be  ascertained  according  to  Art.  381 — 
taking  the  vertical  distance  across  the  falling-mould  at  the  joints, 
and  placing  it  down  from  the  two  outside  heights  in  Fig .  284  or 
285.  After  bevilling  the  plank,  the  moulds  are  applied  as  at  Art. 
383 — applying  the  pitch-board  on  the  bevilled  instead  of  a  square 
edge,  and  placing  the  tips  of  the  mould  so  that  they  will  bear  the 
same  relation  to  the  edge  of  the  plank,  as  they  do  to  the  line,  j  l , 
in  Fig.  284  or  285. 

W 


390. _ To  apply  the  moulds  without  bevilling  the  plank. 

Make  w  p,  {Fig.  291,)  equal  to  w  p  at  Fig.  289,  and  the  angle, 
bed ,  equal  to  bj  l  in  Fig.  284  ;  make  p  a  equal  to  the  thick¬ 
ness  of  the  plank,  as  w  a  in  Fig.  289,  and  from  a  draw  a  o.  pa¬ 
rallel  to  w  d  ;  from  c,  draw  c  e,  at  right  angles  to  w  d,  and  join  e 


240 


AMERICAN  HOUSE-CARPENTER. 


and  b  ;  then  the  angle,  b  e  o,  on  a  square  edge  of  the  plank,  hav¬ 
ing  a  line  on  the  upper  face  at  the  distance,  p  a,  in  Fig.  289,  at 
which  to  apply  the  tips  of  the  mould — will  answer  the  same  pur¬ 
pose  as  bevilling  the  edge. 

If  the  bevilled  edge  of  the  plank,  which  reaches  from  p  to  w, 
is  supposed  to  be  in  the  plane  of  the  paper,  and  the  point,  a,  to 
be  above  the  plane  of  the  paper  as  much  as  a ,  in  Fig.  289,  is  dis¬ 
tant  from  the  line,  wp  ;  and  the  plank  to  be  revolved  on  p  b  as 
an  axis  until  the  line,  p  w ,  falls  below  the  plane  of  the  paper,  and 
the  line,  p  a,  arrives  in  it ;  then,  it  is  evident  that  the  point,  c, 
will  fall,  in  the  line,  c  e,  until  it  lies  directly  behind  the  point,  e, 
and  the  line,  b  c,  will  lie  directly  behind  b  e. 


k 


391. —  To  find  the  bevils  for  splayed  work.  The  principle 
employed  in  the  last  figure  is  one  that  will  serve  to  find  the  bevils 
for  splayed  work — such  as  hoppers,  bread-trays,  (fee. — and  a  way 
of  applying  it  to  that  purpose  had  better,  perhaps,  be  introduced 
in  this  connection.  In  Fig.  292,  a  b  c  is  the  angle  at  which  the 
work  is  splayed,  and  b  d ,  on  the  upper  edge  of  the  board,  is  at 
right  angles  to  a  b  ;  make  the  angle,  fgj,  equal  to  a  b  c,  and 
from/,  draw/ h,  parallel  to  e  a  ;  from  b,  draw  b  o,  at  right  an¬ 
gles  to  a  b  ;  through  o,  draw  i  e,  parallel  to  c  b,  and  join  e  and 
d  ;  then  the  angle,  a  e  d,  will  be  the  proper  bevil  for  the  ends  from 
the  inside,  or  k  d  e  from  the  outside.  If  a  mitre-joint  is  re- 


Stairs 


24  i 


qiliredj  set  f  g,  the  thickness  of  the  stuff  on  the  level,  from  e  to 
m,  and  join  m  and  d ;  then  k  d  m  will  be  the  proper  bevil  for  a 
mitre-joint. 

If  the  upper  edges  of  the  splayed  work  is  to  be  bevilled,  so  as 
to  be  horizontal  when  the  work  is  placed  in  its  proper  position) 
f  g  j,  being  the  same  as  a  b  c,  will  be  the  proper  bevil  for  that 
purpose.  Suppose,  therefore,  that  a  piece  indicated  by  the  lines, 
k  g,  g  f  and  f  h,  were  taken  off ;  then  a  line  drawn  upon  the 
bevilled  surface  from  d,  at  right  angles  to  Ic  d,  would  show  the 
true  position  of  the  joint,  because  it  would  be  in  the  direction  of 
the  board  for  the  other  side ;  but  a  line  so  drawn  would  pass 
through  the  point,  o, — thus  proving  the  principle  correct.  So,  if 
a  line  were  drawn  upon  the  bevilled  surface  from  d,  at  an  angle 
of  45  degrees  to  k  d,  it  would  pass  through  the  point,  n. 

392. — Another  method  for  face-moulds.  It  will  be  seen  by 
reference  to  Art.  388,  that  the  principal  object  had  in  view  in  the 
preparatory  process  of  finding  a  face-mould,  is  to  ascertain  upon  it 
the  direction  of  a  horizontal  line.  This  can  be  found  by  a  method 
different  from  any  previously  proposed  ;  and  as  it  requires  fewer 
lines,  and  admits  of  less  complication,  it  is  probably  to  be  preferred* 
It  can  be  best  introduced,  perhaps,  by  the  following  explanation : 

In  Fig.  293,  j  d  represents  a  prism  standing  upon  a  level  base, 
b  d,  its  upper  surface  forming  an  acute  angle  with  the  face, 
b  l,  as  at  Fig.  287.  Extend  the  base  line,  b  c,  and  the  raking 
line,  j  l ,  to  meet  at  f;  also,  extend  e  d  and  g  a,  to  meet  at  k; 
from  f  through  k,  draw  /  m.  If  we  suppose  the  prism  to  stand 
upon  a  level  floor,  o  f  m,  and  the  plane,  j  gal,  to  be  extended 
to  meet  that  floor,  then  it  Will  be  obvious  that  the  intersection 
between  that  plane  and  the  plane  of  the  floor  would  be  in  the  line, 
f  k  ;  and  the  line,  /  k ,  being  in  the  plane  of  the  floor,  and  also  in 
the  inclined  plane,  j  g  k /,  any  line  made  in  the  plane,  j  g  lc  f 
parallel  to  /  k,  must  be  a  level  line.  By  finding  the  position  of  a 
perpendicular  plane,  at  right  angles  to  the  raking  plane,  j  f  k  g, 
we  shall  greatly  shorten  the  process  for  obtaining  ordinates. 

31 


242 


AMERICAN  HOUSE-CARPENTER* 


This  may  be  done  thus  :  from  f  draw /  o,  at  right  angles  to  fm  ; 
extend  e  b  to  o,  and  g  j,  to  t ;  from  o,  draw  o  t,  at  right  angles  to 
of,  and  join  t  and/;  then  t  of  will  be  a  perpendicular  plane,  at 
right  angles  to  the  inclined  plane,  t  g  k  f ;  because  the  base  of 
the  former,  o  f  is  at  right  angles  to  the  base  of  the  latter,/  k ,  both 
these  lines  being  in  the  same  plane.  From  b,  draw  b  p,  at  right 
angles  to  o  f  or  parallel  to/m  ;  from/?,  draw  p  q ,  at  right  angles 
to  o  /,  and  from  q,  draw  a  line  on  the  upper  plane,  parallel  to  fm, 
or  at  right  angles  to  t  f ;  then  this  line  will  obviously  be  drawn 
to  the  point,  /  and  the  line,  qj,  be  equal  to  p  b.  Proceed,  in  the 
same  way,  from  the  points,  6'  and  c,  to  find  x  and  l. 

Now,  to  apply  the  principle  here  explained,  let  the  curve,  b  s  c, 
{Fig.  294,)  be  the  base  of  a  cylindric  segment,  and  let  it  be  re¬ 
quired  to  find  the  shape  of  a  section  of  this  segment,  cut  by  a 
plane  passing  through  three  given  points  in  its  curved  surface : 
one  perpendicularly  over  6,  at  the  height,  bj;  one  perpendicu¬ 
larly  over  s ,  at  the  height,  s  x  ;  and  the  other  over  c,  at  the  height, 
c  l — these  lines  being  drawn  at  right  angles  to  the  chord  of  the 
base,  b  c.  From /  through  l,  draw  a  line  to  meet  the  chord  line 
extended  to/;  from  s,  draw  5  k,  parallel  to  b  f  and  from  x , 
draw  x  k,  parallel  toy/;  from  /  through  k,  draw  fm;  then  f  m 
will  be  the  intersecting  line  of  the  plane  of  the  section  with  the 


STAIRS. 


243 


Fig.  294. 


plane  of  the  base.  This  line  can  be  proved  to  be  the  intersection 
of  these  planes  in  another  way  ;  from  b,  through  s ,  and  from  j, 
through  x,  draw  lines  meeting  at  m  ;  then  the  point,  m,  will  be 
in  the  intersecting  line,  as  is  shown  in  the  figure,  and  also  at 
Fig.  293. 

From /  draw  fp,  at  right  angles  to  /  m  ;  from  b  and  c,  and 
from  as  many  other  points  as  is  thought  necessary,  draw  ordinates, 
parallel  to  f  m;  make  p  q  equal  to  b  j,  and  join  q  and/;  from 
the  points  at  which  the  ordinates  meet  the  line,  qf,  draw  others 
at  right  angles  to  q  f;  make  each  ordinate  at  A  equal  to  its  cor¬ 
responding  ordinate  at  C ,  and  trace  the  curve,  gii  i,  through  the 
points  thus  found. 

Now  it  may  be  observed  that  A  is  the  plane  of  the  section,  B 
the  plane  of  the  segment,  corresponding  to  the  plane,  q  pf:  of 
Fig.  293,  and  C  is  the  plane  of  the  base.  To  give  these  planes 
their  proper  position,  let  A  be  turned  on  qf  as  an  axis  until  it 


244 


AMERICAN  HOUSE-CARPENTER. 


stands  perpendicularly  over  the  line,  qf  and  at  right  angles  to 
the  plane,  B  ;  then,  while  A  and  B  are  fixed  at  right  angles,  let 
B  be  turned  on  the  line,  p  f  as  an  axis  until  it  stands  perpendicu¬ 
larly  over  p  f  and  at  right  angles  to  the  plane,  C ;  then  the  plane, 
A,  will  lie  over  the  plane,  C,  with  the  several  lines  on  one  corres¬ 
ponding  to  those  on  the  other  ;  the  point,  i ,  resting  at  l,  the  point, 
?i,  at  x,  and  g  at  j  ;  and  the  curve,  g  n  i,  lying  perpendicularly 
over  b  s  c — as  was  required.  If  we  suppose  the  cylinder  to  be 
cut  by  a  level  plane  passing  through  the  point,  l ,  (as  is  done  in 
finding  a  face-mould,)  it  will  be  obvious  that  lines  corresponding 
to  qf  and  p  f  would  meet  in  l ;  and  the  plane  of  the  section,  A, 
the  plane  of  the  segment,  B,  and  the  plane  of  the  base,  C,  would 
all  meet  in  that  point. 

393. —  To  find  the  face-mould  for  a  hand-rail  according  to 
the  principles  explained  in  the  previous  article.  In  Fig.  295, 
ae  cf  is  the  plan  of  a  hand-rail  over  a  quarter  of  a  cylinder ;  and 
in  Fig.  296,  abed  is  the  falling-mould ;  /  e  being  equal  to  the 
stretch-out  of  a  df  in  Fig.  295.  From  c,  draw  c  h,  parallel  to 
ef;  bisect  h  c  in  i ,  and  find  a  point,  as  b,  in  the  arc,  df  {Fig. 
295.)  corresponding  to  i  in  the  line,  h  c ;  from  i,  {Fig.  296,)  to 
the  top  of  the  falling-mould,  draw  i  j ,  at  right  angles  to  he;  at  Fig. 
295,  from  c,  through  b,  draw  c  g,  and  from  b  and  c,  draw  b  j  and 
c  k,  at  right  angles  to  g  c  ;  make  c  k  equal  to  h  g  at  Fig.  296, 
and  b  j  equal  to  i  j  at  that  figure  ;  from  k,  through  j ,  draw  k  g, 
and  fromg’,  through  a,  draw  gp  ;  then  gp  will  be  the  intersecting 
line,  corresponding  to/ m  in  Fig.  293  and  294  ;  through  <?,  draw 
p  6,  at  right  angles  to  g  p,  and  from  c,  draw  c  q.  parallel  to  g  p  ; 
make  r  q  equal  to  li  g  at  Fig.  296  ;  joinp  and  q,  and  proceed  as 
in  the  previous  examples  to  find  the  face-mould,  A.  The  joint 
of  the  face-mould,  u  v ,  will  be  more  accurately  determined  by 
finding  the  projection  of  the  centre  of  the  plan,  o,  as  at  w  ; 
joining  s  and  w,  and  drawing  u  v,  parallel  to  s  w. 

It  may  be  noticed  that  c  k  and  b  j  are  not  of  a  length  corres¬ 
ponding  to  the  above  directions  :  they  are  but  I  the  length  given. 


STAIRS. 


245 


246 


AMERICAN  HOUSE-CARPENTER, 


Ji 


Fig.  296. 


The  object  of  drawing  these  lines  is  to  find  the  point,  g,  and  that 
can  be  done  by  taking  any  proportional  parts  of  the  lines  given, 
as  well  as  by  taking  the  whole  lines.  For  instance,  supposing  c 
k  and  b  j  to  be  the  full  length  of  the  given  lines,  bisect  one  in  i 
and  the  other  in  m ;  then  a  line  drawn  from  m,  through  i,  will 
give  the  point,  g:  as  was  required.  The  point,  g,  may  also  be 


STAIRS. 


247 


obtained  thus :  at  Fig.  296,  make  h  l  equal  to  c  b  in  Fig.  295  ; 
from  l,  draw  l  k,  at  right  angles  to  h  c  ;  from  j,  drawy  k,  parallel 
to  he;  from  g,  through  k,  draw  g  n  ;  at  Fig.  295,  make  b  g 
equal  to  l  n  in  Fig.  296  ;  then  g  will  be  the  point  required. 

The  reason  why  the  points,  a,  b  and  c,  in  the  plan  of  the  rail  at 
Fig.  295,  are  taken  for  resting  points  instead  of  e ,  i  and/,  is  this  : 
the  top  of  the  rail  being  level,  it  is  evident  that  the  points,  a  and  e, 
in  the  section  a  e,  are  of  the  same  height ;  also  that  the  point,  i,  is  of 
the  same  height  as  6,  and  c  as  /.  Now,  if  a  is  taken  for  a  point 
in  the  inclined  plane  rising  from  the  line  g  p,  e  must  be  below 
that  plane  ;  if  b  is  taken  for  a  point  in  that  plane,  i  must  be  below 
it ;  and  if  c  is  in  the  plane,/  must  be  below  it.  The  rule,  then, 
for  taking  these  points,  is  to  take  in  each  section  the  one  that  is 
nearest  to  the  line,  g  p.  Sometimes  the  line  of  intersection,  g  p , 
happens  to  come  almost  in  the  direction  of  the  line,  e  r  :  in  such 
case,  after  finding  the  line,  see  if  the  points  from  which  the 
heights  were  taken  agree  with  the  above  rule  ;  if  the  heights 
were  taken  at  the  wrong  points,  take  them  according  to  the  rule 
above,  and  then  find  the  true  line  of  intersection,  which  will  not 
vary  much  from  the  one  already  found. 


394. —  To  apply  the  face-mould  thus  found  to  the  plank. 
The  face-mould,  when  obtained  by  this  method,  is  to  be  applied 
to  a' square-edged  plank,  as  directed  at  Art.  383,  with  this  differ¬ 
ence  :  instead  of  applying  both  tips  of  the  mould  to  the  edge  of 


248 


AMERICAN  HOUSE-CARPENTER. 


the  plank,  one  of  them  is  to  be  set  as  far  from  the  edge  of  the 
plank,  as  x ,  in  Fig-.  295,  is  from  the  chord  of  the  section  p  q — as 
is  shown  at  Fig.  29 7.  A,  in  this  figure,  is  the  mould  applied  on 
the  upper  side  of  the  plank,  B ,  the  edge  of  the  plank,  and  C,  the 
mould  applied  on  the  under  side  ;  a  b  and  c  cL  being  made  equal 
to  q  x  in  Fig.  295,  and  the  angle,  e  a  c,  on  the  edge,  equal  to  the 
angle,  p  q  r,  at  Fig.  295.  In  order  to  avoid  a  waste  of  stuff,  it 
would  be  advisable  to  apply  the  tips  of  the  mould,  e  and  b ,  im¬ 
mediately  at  the  edge  of  the  plank.  To  do  this,  suppose  the 
moulds  to  be  applied  as  shown  in  the  figure ;  then  let  A  be  re¬ 
volved  upon  e  until  the  point,  b,  arrives  at  g,  causing  the  line,  e  b , 
to  coincide  with  e  g :  the  mould  upon  the  under  side  of  the 
plank  must  now  be  revolved  upon  a  point  that  is  perpendicularly 
beneath  e,  as  /;  from/,  draw  f  h,  parallel  to  i  d,  and  from  d, 
draw  d  h,  at  right  angles  to  id;  then  revolve  the  mould,  C,  upon 
/  until  the  point,  Zt,  arrives  at  /  causing  the  line,/  h ,  to  coincide 
with//  and  the  line,  i  d,  to  coincide  with  k  l ;  then  the  tips  of 
the  mould  will  be  at  k  and  l. 

The  rule  for  doing  this,  then,  will  be  as  follows  :  make  the  an¬ 
gle,  if  k,  equal  to  the  angle  q  v  x,  at  Fig.  295  ;  make /  k  equal 
to  fi,  and  through  k,  draw  k  l,  parallel  to  ij;  then  apply  the 
corner  of  the  mould,  i,  at  k:  and  the  other  corner  d,  at  the  line,  k  L 

The  thickness  of  stuff  is  found  as  at  Art.  381. 

395. —  To  regulate  the  application  of  the  falling-mould. 
Obtain,  on  the  line,  h  c,  {Fig.  296,)  the  several  points,  r,  q,  p,  l 
and  m,  corresponding  to  the  points,  b\  a\  z ,  y,  &c.,  at  Fig.  295  ; 
from  r  q  p,  &c.,  draw  the  lines,  r  t ,  q  u,p  v ,  &c.,  at  right  angles 
to  h  c;  make  h  s ,  r  t,  q  u,  &c.,  respectively  equal  to  6  c2,  r  q,  5 
d2,  &c.,  at  Fig.  295 ;  through  the  points  thus  found,  trace  the 
curve,  s  w  c.  Then  get  out  the  piece,  g  s  c,  attached  to  the  fall¬ 
ing-mould  at  several  places  along  its  length,  as  at  z,  z,  z ,  &c. 
In  applying  the  falling-mould  with  this  strip  thus  attached,  the 
edge,  sw  c,  will  coincide  with  the  upper  surface  of  the  rail  piece 


STAIRS.  249 

# 

before  it  is  squared  ;  and  thus  show  the  proper  position  of  the  fall¬ 
ing-mould  along  its  whole  length.  (See  Art.  403.) 

SCROLLS  FOR  HAND-RAILS. 

396. —  General  rule  for  finding  the  size  and  position  of  the 
regulating  square.  The  breadth  which  the  scroll  is  to  occupy, 
the  number  of  its  revolutions,  and  the  relative  size  of  the  regula¬ 
ting  square  to  the  eye  of  the  scroll,  being  given,  multiply  the 
number  of  revolutions  by  4,  and  to  the  product  add  the  number 
of  times  a  side  of  the  square  is  contained  in  the  diameter  of  the 
eye,  and  the  sum  will  be  the  number  of  equal  parts  into  which 
the  breadth  is  to  be  divided.  Make  a  side  of  the  regulating 
square  equal  to  one  of  these  parts.  To  the  breadth  of  the  scroll 
add  one  of  the  parts  thus  found,  and  half  the  sum  will  be  the 
length  of  the  longest  ordinate. 


2  1 


397. —  To  find  the  proper  centres  in  the  regulating  square. 
Let  a  2  1  b,  (Fig.  298,)  be  the  size  of  a  regulating  square,  found 
according  to  the  previous  rule,  the  required  number  of  revolu¬ 
tions  being  If.  Divide  two  adjacent  sides,  as  a  2  and  2  1,  into 
as  many  equal  parts  as  there  are  quarters  in  the  number  of  revo¬ 
lutions,  as  seven  ;  from  those  points  of  division,  draw  lines  across 
the  square,  at  right  angles  to  the  lines  divided ;  then,  1  being  the 
first  centre,  2,  3,  4,  5,  6  and  7,  are  the  centres  for  the  other  quar¬ 
ters,  and  8  is  the  centre  for  the  eye  ;  the  heavy  lines  that  deter- 

32 


250 


AMERICAN  HOUSE-CARPENTER. 


mine  these  centres  being  each  one  part  less  in  length  than  its  pre¬ 
ceding  line. 


hi 


398. —  To  describe  the  scroll  for  a  hand-rail  over  a  curtail 
step.  Let  a  b,  (Fig.  299,)  be  the  given  breadth,  If  the  given 
number  of  revolutions,  and  let  the  relative  size  of  the  regulating 
square  to  the  eye  be  £  of  the  diameter  of  the  eye.  Then,  by  the 
rule,  If  multiplied  by  4  gives  7,  and  3,  the  number  of  times  a 
side  of  the  square  is  contained  in  the  eye,  being  added,  the  sum 
is  10.  Divide  a  b ,  therefore,  into  10  equal  parts,  and  set  one  from 
b  to  c  ;  bisect  acme;  then  a  e  will  be  the  length  of  the  longest 
ordinate,  (1  d  or  1  e.)  From  a,  draw  a  d,  from  e,  draw  e  1,  and 
from  6,  draw  bf  all  at  right  angles  to  a  b  ;  make  e  1  equal  to  e 
a,  and  through  1,  draw  1  d,  parallel  to  a  b  ;  set  b  c  from  1  to  2, 
and  upon  1  2,  complete  the  regulating  square ;  divide  this  square 
as  at  Fig.  298  ;  then  describe  the  arcs  that  compose  the  scroll,  as 
follows:  upon  1,  describe  d  e;  upon  2,  describe  e  f;  upon  3, 
describe  /  g  ;  upon  4,  describe  g  hy  &c. ;  make  d  l  equal  to  the 


STAIRS. 


251 


width  of  the  rail,  and  upon  1,  describe  Im  ;  upon  2,  aescribe  m 
n,  &c. ;  describe  the  eye  upon  8,  and  the  scroll  is  completed. 

399.  —  To  describe  the  scroll  for  a  curtail  step.  Bisect  d  l, 
(Tig.  299,)  in  o,  and  make  o  v  equal  to  £  of  the  diameter  of  a 
baluster ;  make  v  w  equal  to  the  projection  of  the  nosing,  and  e 
x  equal  to  w  l;  upon  1,  describe  w  y ,  and  upon  2,  describe  y  z  ; 
also  upon  2,  describe  x  i  ;  upon  3,  describe  ij ,  and  so  around  to 
z  ;  and  the  scroll  for  the  step  will  be  completed. 

400.  —  To  determine  the  position  of  the  balusters  under  the 
scroll.  Bisect  d  l,  (Fig.  299,)  in  o,  and  upon  1,  with  1  o  for  ra¬ 
dius,  describe  the  circle,  or  u;  set  the  baluster  at  p  fair  with  the 
face  of  the  second  riser,  c2,  and  from  p,  with  half  the  tread  in  the 
dividers,  space  off  as  at  o,  q ,  r,  s,  t ,  u ,  <fec.,  as  far  as  q2 ;  upon  2, 
3,  4  and  5,  describe  the  centre-line  of  the  rail  around  to  the  eye 
of  the  scroll ;  from  the  points  of  division  in  the  circle,  o  r  u,  draw 
lines  to  the  centre-line  of  the  rail,  tending  to  the  centre  of  the 
eye,  8;  then,  the  intersection  of  these  radiating  lines  with  the 
centre-line  of  the  rail,  will  determine  the  position  of  the  balusters, 
as  shown  in  the  figure. 

j 


401.—  To  obtain  the  falling-mould  for  the  raking  part  of  the 
scroll.  Tangical  to  the  rail  at  h,  (Fig.  299,)  draw  h  k ,  parallel  to  d 
a;  then  k  a 2  will  be  the  joint  between  the  twist  and  the  other  part 
of  the  scroll.  Make  d  e2  equal  to  the  stretch-out  of  de,  and  upon  d 


252 


AMERICAN  HOUSE-CARPENTER. 


e )  find  the  position  of  the  point,  k}  as  at  k* ;  at  Fig.  300,  make  e  d 
equal  to  e2  d  in  Fig.  299,  and  d  c  equal  to  d  c 2  in  that  figure  ; 
from  c,  draw,  c  a,  at  right  angles  to  e  c,  and  equal  to  one  rise  ; 
make  c  b  equal  to  one  tread,  and  from  5,  through  a,  draw  bj; 
bisect  a  c  in  /,  and  through  l,  draw  m  q ,  parallel  to  e  h  ;  m  q  is 
the  height  of  the  level  part  of  a  scroll,  which  should  always  be 
about  3J  feet  from  the  floor ;  ease  off  the  angle,  m  f j,  according 
to  Art.  89,  and  draw  g  w  n,  parallel  to  m  x  j ,  and  at  a  distance 
equal  to  the  thickness  of  the  rail ;  at  a  convenient  place  for  the 
joint,  as  i,  draw  in,  at  right  angles  to  b  j  ;  through  n,  draw  j  k , 
at  right  angles  to  c  h  ;  make  d  k  equal  to  d  Id  in  Fig.  299,  and 
from  k,  draw  k  o,  at  right  angles  to  e  h  ;  at  Fig.  299,  make  d 
Id  equal  to  d  h  in  Fig.  300,  and  draw  Id  b2,  at  right  angles  to  d 
Id  ;  then  k  cd  and  Id  d  will  be  the  position  of  the  joints  on  the 
plan,  and  at  Fig.  300,  o  p  and  i  n,  their  position  on  the  falling- 
mould  ;  and  pox  n,  {Fig.  300,)  will  be  the  falling-mould  re¬ 
quired. 


r  n  o  p  q  ’  l  b 


402. —  To  describe  the  face-mould.  At  Fig.  299,  from  k,  draw 
k  r2,  at  right  angles  to  r2  d  ;  at  Fig.  300,  make'/i  r  equal  to  hd  r* 
in  Fig.  299,  and  from  r,  draw  r  s ,  at  right  angles  to  r  h  ;  from 
the  intersection  of  r  s  with  the  level  line,  m  q,  through  i,  draw  s 
t ;  at  Fig.  299,  make  Id  b'2  equal  to  q  t  in  Fig.  300,  and  join  td 
and  r2 ;  from  a2,  and  from  as  many  other  points  in  the  arcs,  a 2 1 
and  k  d,  as  is  thought  necessary,  draw  ordinates  to  r 3  d,  at  right 
angles  to  the  latter ;  make  r  b,  {Fig.  301,)  equal  in  its  length  and 
in  its  divisions  to  the  line,  r2  b2,  in  Fig.  299  ;  from  r,  n,  o ,  p,  q 


STAIRS. 


253 


and  l}  draw  the  lines,  r  k,  n  d:  o  a,  p  e,  qf  and  l  c,  at  right  an¬ 


gles  to  r  b,  and  equal  to  r7  k ,  d2  s2,  /2  a2,  &c.,  in  Fig-.  299 ; 
through  the  points  thus  found,  trace  the  curves,  k  l  and  a  c,  and 
complete  the  face-mould,  as  shown  in  the  figure.  This  mould  is 
to  be  applied  to  a  square-edged  plank,  with  the  edge,  l  b:  parallel 
to  the  edge  of  the  plank.  The  rake  lines  upon  the  edge  of  the 
plank  are  to  be  made  to  correspond  to  the  angle,  s  t  h,  in  Fig. 
300.  The  thickness  of  stuff  required  for  this  mould  is  shown  at 
Fig.  300,  between  the  lines  5  t  and  u  v — u  v  being  drawn  pa¬ 
rallel  to  s  t. 

403. — All  the  previous  examples  given  for  finding  face-moulds 
over  winders,  are  intended  for  moulded  rails.  For  round  rails, 
the  same  process  is  to  be  followed  with  this  difference :  instead 
of  working  from  the  sides  of  the  rail,  work  from  a  centre-line. 
After  finding  the  projection  of  that  line  upon  the  upper  plane, 
describe  circles  upon  it,  as  at  Fig.  262,  and  trace  the  sides  of  the 
moulds  by  the  points  so  found.  The  thickness  of  stuff  for  the 
twists  of  a  round  rail,  is  the  same  #s  for  the  straight ;  and  the 


twists  are  to  be  sawed  square  through. 


h  f  v  k 


254 


AMERICAN  HOUSE-CARPENTER. 


404. —  To  ascertain  the  form  of  the  newel-cap  from  a  section 
of  the  rail.  Draw  a  h ,  {Fig.  302,)  through  the  widest  part  of 
the  given  section,  and  parallel  to  erf;  bisect  a  b  in  e,  and  through 
a,  e  and  b.  draw  hi,  f  g  and  kj,  at  right  angles  to  a  b  ;  at  a  con¬ 
venient  place  on  the  line,/ g ,  as  o,  with  a  radius  equal  to  half 
the  width  of  the  cap,  describe  the  circle,  i  j  g  ;  make  r  l  equal 
to  e  b  or  e  a  ;  join  l  and/  also  l  and  i;  from  the  curve,  /  b,  to 
the  line,  l  j ,  draw  as  many  ordinates  as  is  thought  necessary, 
parallel  to  f  g;  from  the  points  at  which  these  ordinates  meet 
the  line,  l  j,  and  upon  the  centre,  o,  describe  arcs  in  continuation  to 
meet  op;  from  n,  t,  x,  &c.,  draw  n  s,  t  u,  &c.,  parallel  to  f  g  ; 
make  n  s,  t  u,  &c.,  equal  to  e  f  w  v,  &c. ;  make  x  y,  & c.,  equal 
to  z  d ,  &c. ;  make  o  2,  o  3,  &c.,  equal  to  o  n,  o  t,  &c. ;  make  2  4 
equal  to  n  s,  and  in  this  way  find  the  length  of  the  lines  crossing 
o  m  ;  through  the  points  thus  found,  describe  the  section  of  the 
newel-cap,  as  shown  in  the  figure. 


APPENDIX 


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GLOSSARY. 


Terms  not  found  here  can  be  found  in  the  lists  of  definitions  in  other  parts  of  this  book* 

or  in  common  dictionaries. 


Abacus. — The  uppermost  member  of  a  capital. 

Abbatoir. — A  slaughter-house. 

Abbey. — The  residence  of  an  abbot  or  abbess. 

Abutment. — That  part  of  a  pier  from  which  the  arch  springs. 

Acanthus. — A  plant  called  in  English,  bear’s-breech.  Its  leaves  ard 
employed  for  decorating  the  Corinthian  and  the  Composite  capitalsi 

Acropolis. — The  highest  part  of  a  city ;  generally  the  citadel. 

Acroleria. — The  small  pedestals  placed  on  the  extremities  and  apex 
of  a  pediment,  originally  intended  as  a  base  for  sculpture. 

Aisle. — Passage  to  and  from  the  pews  of  .a  church.  In  Gothic  ar¬ 
chitecture,  the  lean-to  wings  on  the  sides  of  the  nave. 

Alcove. — Part  of  a  chamber  separated  by  an  estrade ,  or  partition  of 
columns.  Recess  with  seats,  &c.,  in  gtrdens. 

Altar. — A  pedestal  whereon  sacrifice  was  offered.  In  modern 
churches,  the  area  within  the  railing  in  front  of  the  pulpit. 

Alto-relievo. — High  relief ;  sculpture  projecting  from  a  surface  so  as 
to  appear  nearly  isolated. 

Amphitheatre. — A  double  theatre,  employed  by  the  ancients  for  the 
exhibition  of  gladiatorial  fights  and  other  shows. 

Ancones. — Trusses  employed  as  an  apparent  support  to  a  cornice 
upon  the  flanks  of  the  architrave. 

Annulet. — A  small  square  moulding  used  to  separate  others  ;  the 
fillets  in  the  Doric  capital  under  the  ovolo,  and  those  which  separate 
the  flutings  of  columns,  are  known  by  this  term. 

Antce. — A  pilaster  attached  to  a  wall. 

Apiary. — A  place  for  keeping  beehives. 

Arabesque. — A  building  after  the  Arabian  style. 

Areostyle. — An  intercolumniation  of  from  four  to  five  diameters. 

'  Arcade — A  series  of  arches. 

Arch. — An  arrangement  of  stones  or  other  material  in  a  curvilinear 
form,  so  as  to  perform  the  office  of  a  lintel  and  carry  superincumbent 
weights. 

Architrave. — That  part  of  the  .entablature  which  rests  upon  the 
capital  of  a  column,  and  is  beneath  the  frieze.  The  casing  and 
mouldings  about  a  door  or  window. 


4 


APPENDIX. 


Archivolt. — The  ceiling  of  a  vaulf :  the  under  surface  of  an  arch. 

Area. — Superficial  measurement.  An  open  space,  below  the  level 
of  the  ground,  in  front  of  basement  windows. 

Arsenal. — A  public  establishment  for  the  deposition  of  arms  and 
warlike  stores. 

Astragal.— k  small  moulding  consisting  of  a  half-round  with  a  fillet 
on  each  side. 

Attic. — A  low  story  erected  over  an  order  of  architecture.  A  low 
additional  story  immediately  under  the  roof  of  a  building. 

Aviary  .—A  place  for  keeping  and  breeding  birds. 

Balcony. — An  open  gallery  projecting  from  the  front  of  a  building. 

Baluster. — A  small  pillar  or  pilaster  supporting  a  rail. 

Balustrade. — A  series  of  balusters  connected  by  a  rail. 

Barge-course. — -That  part  of  the  covering  which  projects  over  the 
gable  of  a  building. 

Base. — The  lowest  pai*t  of  a  wall,  column,  &c. 

Basement-story. — That  which  is  immediately  under  the  principal 
story,  and  included  within  the  foundation  of  the  building. 

Basso-relievo. — Low  relief ;  sculptured  figures  projecting  from  a 
surface  one-half  their  thickness  or  less.  See  Alto-relievo. 

Battering. — See  Talus. 

Battlement. — Indentations  on  the  top  of  a  wall  or  parapet. 

Bay-window. — A  window  projecting  in  two  or  more  planes,  and'  not 
forming  the  segment  of  a  circle. 

Bazaar. — A  species  of  mart  or  exchange  for  the  sale  of  various  ar¬ 
ticles  of  merchandise. 

Bead. — A  circular  moulding. 

Bed-mouldings. — Those  mouldings  which  are  between  the  corona 
and  the  frieze.  » 

Belfry.— That  part  of  a  steeple  in  which  the  bells  are  hung :  an¬ 
ciently  called  campanile. 

Belvedere. — An  ornamental  turret  or  observatory  commanding  at 
pleasant  prospect. 

Bow-windoic. — A  window  projecting  in  curved  lines. 

Bressummer . — Abeam  or  iron  tie  supporting  a  wall  over  a  gateway 
or  other  opening. 

Brick -nogging. — The  brickwork  between  studs  of  partitions. 

Buttress. — A  projection  from  a  wall  to  give  additional  strength. 

Cubic. — A  cylindrical  moulding  placed  in  flutes  at  the  lower  part  of 
the  column. 

Camber. — To  give  a  convexity  to  the  upper  surface  of  a  beam. 

Campanile. — A  tower  for  the  reception  of  bells,  usually,  in  Italy, 
separated  from  the  church. 

Canopy. — An  ornamental  covering  over  a  seat  of  state. 

Cantalivers. — The  ends  of  rafters  under  a  projecting  roof.  Pieces 
of  wood  or  stone  supporting  the  eaves. 

Capital. — The  uppermost  part  of  a  column  included  between  the 
shaft  and  the  architrave. 


APPENDIX.  5 

Caravansera.— In  the  East,  a  large  public  building  for  the  reception 
of  travellers  by  caravans  in  the  desert. 

Carpentry. — (From  the  Latin,  carpentum,  carved  wood.)  That  de¬ 
partment  of  science  and  art  which  treats  of  the  disposition,  the  con¬ 
struction  and  the  relative  strength  of  timber.  Tin  first  is  called  de¬ 
scriptive,  the  second  constructive,  and  the  last  mechanical  carpentry. 

Caryatides . — Figures  of  women  used  instead  of  columns  to  support 
an  entablature. 

Casino. — A  small  country-house. 

Castellated. — Built  with  battlements  and  turrets  in  imitation  of  an¬ 
cient  castles. 

Castle. — A  building  fortified  for  military  defence.  A  house  with 
towers,  usually  encompassed  with  walls  and  moats,  and  having  a  don¬ 
jon,  or  keep,  in  the  centre. 

Catacombs. — Subterraneous  places  for  burying  the  dead. 

Cathedral. — The  principal  church  of  a  province  or  diocese,  wherein 
the  throne  of  the  archbishop  or  bishop  is  placed. 

Cavetto. — A  concave  moulding  comprising  the  quadrant  of  a  circle. 

Cemetery. — An  edifice  or  area  where  the  dead  are  interred. 

Cenotaph. — A  monument  erected  to  the  memory  of  a  person  buried 
in  another  place. 

Centring. — The  temporary  woodwork,  or  framing,  whereon  any 
vaulted  work  is  constructed. 

Cesspool. — A  well  under  a  drain  or  pavement  to  receive  the  waste- 
water  and  sediment. 

Chamfer. — The  bevilled  edge  of  any  thing  originally  right-angled. 

Chancel. — That  part  of  a  Gothic  church  in  which  the  altar  is  placed. 

Chantry. — A  little  chapel  in  ancient  churches,  with  an  endowment 
for  one  or  more  priests  to  say  mass  for  the  relief  of  souls  out  of  purga¬ 
tory. 

Chapel. — A  building  for  religious  worship,  erected  separately  from 
a  church,  and  served  by  a  chaplain. 

Chaplet. — -A  moulding  carved  into  beads,  olives,  &c. 

Cincture. — The  ring,  listel,  or  fillet,  at  the  top  and  bottom  of  a  co¬ 
lumn,  which  divides  the  shaft  of  the  column  from  its  capital  and  base. 

Circus. — A  straight,  long,  narrow  building  used  by  the  Romans  for 
the  exhibition  of  public  spectacles  and  chariot  races.  At  the  present 
day,  a  building  enclosing  an  arena  for  the  exhibition  of  feats  of  horse¬ 
manship. 

Clerestory . — The  upper  part  of  the  nave  of  a  church  above  the 
roofs  of  the  aisles. 

Cloister. — The  square  space  attached  to  a  regular  monastery  or 
large  church,  having  a  peristyle  or  ambulatory  around  it,  covered  with 
a  range  of  buildings. 

Coffer-dam. — A  case  of  piling,  water-tight,  fixed  in  the  bed  of  a 
river,  for  the  purpose  of  excluding  the  water  while  any  work,  such  as 
a  wharf,  wall,  or  the  pier  of  a  bridge,  is  carried  up. 

Collar-beam. — A  horizontal  beam  framed  between  two  principal 
rafters  above  the  tie-beam. 

Collonade. — A  range  of  columns. 

Columbarium. — A  pigeon-house. 


6 


APPENDIX. 


Column.-T- A  vertical,  cylindrical  support  under  the  entablature  of 
an  order. 

Common-rafters. — The  same  as  jack-rafters,  which  see 

Conduit. — A  long,  narrow,  walled  passage  underground,  for  secret 
pommunication  between  different  apartments.  A  canal  or  pipe  for  the 
conveyance  of  water. 

Conservatory.—: A  building  for  preserving  curious  and  rare  exotic 
plants. 

Consoles. — The  same  as  ancones,  which  see. 

Contour. — The  external  lines  which  bound  and  terminate  a  figure. 

Convent. — A  building  for  the  reception  of  a  society  of  religious  per¬ 
sons. 

Coping. — Stones  laid  on  the  top  of  a  wall  to  defend  it  from  the 
weather. 

Corbels.— Stones  or  timbers  fixed  in  a  wall  to  sustain  the  timbers  of 
a  floor  or  roof. 

Cornice. — Any  moulded  projection  which  crowns  or  finishes  the 
part  to  which  it  is  affixed. 

Corona. — That  part  of  a  cornice  which  is  between  the  crown- 
jnoulding  and  the  bed-mouldings. 

Cornucopia. — The  horn  of  plenty. 

Corridor.— An  open  gallery  or  communication  to  the  different  apart¬ 
ments  of  a  house. 

Cove.— A.  concave  moulding. 

Cripple-rafters. — The  short  rafters  which  are  spiked  to  the  hip-rafter 
of  a  roof. 

Crockets. — In  Gothic  architecture,  the  ornaments  placed  along  the 
angles  of  pediments,  pinnacles,  &c. 

Crosettes. — The  same  as  ancones,  which  see. 

Crypt. — -The  under  or  hidden  part  of  a  building. 

Culvert. — -An  arched  channel  of  masonry  or  brickwork,  built  be¬ 
neath  the  bed  of  a  canal  for  the  purpose  of  conducting  water  under  it. 
Any  arched  channel  for  water  underground. 

Cupola. — A  small  building  on  the  top  of  a  dome. 

Curtail-step. — A  step  Avith  a  spiral  end,  usually  the  first  of  the  flight. 

Cusps.- — The  pendents  of  a  pointed  arch. 

Cyma,—kn  ogee.  There  are  two  kinds ;  the  cyma-recta,  having 
the  upper  part  concave  and  the  lower  convex,  and  the  cyma-rcversa, 
with  the  upper  part  convex  and  the  lower  concave. 

Dado. — The  die,  or  part  between  the  base  and  cornice  of  a  pedestal. 

Dairy.— An  apartment  or  building  for  the  preservation  of  milk,  and 
the  manufacture  of  it  into  butter,  cheese,  &c. 

Dead-shoar. — A  piece  of  timber  or  stone  stood  vertically  in  brick¬ 
work,  to  support  a  superincumbent  weight  until  the  brickwork  which 
is  to  .carry  it  has  set  or  become  hard. 

Decaslyle. — A  building  having  ten  columns  in  front. 

Dentils.— (From  the  Latin,  dentes,  teeth.)  Small  rectangular  blocks 
used  in  the  bed-mouldings  of  some  of  the  orders. 

Diastyle. — An  intercolumniation  of  three,  or,  as  some  say,  four 
diameters. 


appendix. 


7 


Die. — -That  part  of  a  pedestal  included  between  the  base  and  the 
cornice  ;  it  is  also  called  a  dado. 

Dodecasiyle. — A  building  having  twelve  columns  in  front. 

Donjon. — A  massive  tower  within  ancient  castles  to  which  the  gar¬ 
rison  might  retreat  in  case  of  necessity. 

Dooks. — A  Scotch  term  given  to  wooden  bricks. 

Dormer. — A  window  placed  on  the  roof  of  a  house,  the  frame  being 
placed  vertically  on  the  rafters. 

Dormitory. — A  sleeping-room. 

Dovecote. — A  building  for  keeping  tame  pigeons.  A  columbarium. 

Echinus. — The  Grecian  ovolo. 

Elevation. — A  geometrical  projection  drawn  on  a  plane  at  right  an¬ 
gles  to  the  horizon. 

Entablature. — That  part  of  an  order  which  is  supported  by  the  co¬ 
lumns  ;  consisting  of  the  architrave,  frieze,  and  cornice. 

Eustyle. — An  intercolumniation  of  two  and  a  quarter  diameters. 

Exchange. — -A  building  in  which  merchants  and  brokers  meet  to 
transact  business. 

Extrados.— The  exterior  curve  of  an  arch. 

Facade. — The  principal  front  of  any  building. 

Face-mould — The  pattern  for  marking  the  plank,  out  of  which  hand- 
railing  is  to  be  cut  for  stairs,  &c. 

Facia ,  or  Fascia. — A  flat  member  like  a  band  or  broad  fillet. 

Falling-mould. — The  mould  applied  to  the  convex,  vertical  surface 
of  the  rail-piece,  in  order  to  form  the  back  and  under  surface  of  the 
rail,  and  finish  the  squaring. 

Festoon. — An  ornament  representing  a  wreath  of  flowers  and  leaves. 

Fillet. — A  narrow  flat  band,  listel,  or  annulet,  used  for  the  separa¬ 
tion  of  one  moulding  from  another,  and  to  give  breadth  and  firmness 
to  the  edges  of  mouldings. 

Flutes. — Upright  channels  on  the  shafts  of  columns. 

Flyers. — Steps  in  a  flight  of  stairs  that  are  parallel  to  each  other. 

Forum. — In  ancient  architecture,  a  public  market ;  also,  a  place 
where  the  common  courts  were  held,  and  law  pleadings  carried  on. 

Foundry. — A  building  in  which  various  metals  are  cast  into  moulds 
or  shapes. 

Frieze. — That  part  of  an  entablature  included  between  the  archi¬ 
trave  and  the  cornice. 

Gable. — The  vertical,  triangular  piece  of  wall  at  the  end  of  a  roof, 
from  the  level  of  the  eaves  to  the  summit. 

Gain. — A  recess  made  to  receive  a  t°non  or  tusk. 

Gallery. — A  common  passage  to  several  rooms  in  an  upper  story. 
A  long  room  for  the  reception  of  pictures.  A  platform  raised  on  co¬ 
lumns,  pilasters,  or  piers. 

Girder. — The  principal  beam  in  a  floor  for  supporting  the  binding 
and  other  joists,  whereby  the  bearing  or  length  is  lessened. 

Glyph. — A  vertical,  sunken  channel.  From  their  number,  those  irt 
the  Doric  order  are  called  triglyphs. 


8 


APPENDIX. 


Granary. — A  building  for  storing  grain,  especially  that  intended  to 
be  kept  for  a  considerable  time. 

Groin. — The  line  formed  by  the  intersection  of  two  arches,  which 
cross  each  other  at  any  angle. 

Gutter. — The  small  cylindrical  pendent  ornaments,  otherwise  called 
drops ,  used  in  the  Doric  order  under  the  triglyphs,  and  also  pendent 
from  the  mutuli  of  the  cornice. 

Gymnasium. — Originally,  a  space  measured  out  and  covered  with 
sand  for  the  exercise  of  athletic  games  •  afterwards,  spacious  buildings 
devoted  to  the  mental  as  well  as  corporeal  instruction  of  youth. 

Hall. — The  first  large  apartment  on  entering  a  house.  The  public 
room  of  a  corporate  body.  A  manor-house. 

Ham. — A  house  or  dwelling-place.  A  street  or  village  :  hence 
Notting  ham,  Bucking  ham,  &c.  Hamlet,  the  diminutive  of  ham,  is  a 
small  street  or  village. 

Helix. — The  small  volute,  or  twist,  under  the  abacus  in  the  Corin¬ 
thian  capital. 

Hem. — The  projecting  spiral  fillet  of  the  Ionic  capital. 

Hexastyle. — A  building  having  six  columns  in  front. 

Hip-rafter. — A  piece  of  timber  placed  at  the  angle  made  by  two  ad¬ 
jacent  inclined  roofs. 

Homestall , — A  mansion-house,  or  seat  in  the  country. 

Hotel,  or  Hostel. — A  large  inn  or  place  of  public  entertainment.  A 
large  house  or  palace. 

Hot-house. — A  glass  building  used  in  gardening. 

Hovel. — An  open  shed. 

Hut. — A  small  cottage  or  hovel  generally  constructed  of  earthy 
materials,  as  strong  loamy  clay,  &c. 

Impost. — The  capital  of  a  pier  or  pilaster  which  supports  an  arch. 

Intaglio. — Sculpture  in  which  the  subject  is  hollowed  out,  so  that 
the  impression  from  it  presents  the  appearance  of  a  bas-relief. 

Inter  calumniation. — The  distance  between  two  columns. 

Intrados. — The  interior  and  lower  curve  of  an  arch. 

Jack-rafters. — Rafters  that  fill  in  between  the  principal  rafters  of  a 
roof ;  called  also  common-rafters. 

Jail. — A  place  of  legal  confinement. 

Jambs. — The  vertical  sides  of  an  aperture. 

Joggle-piece. — A  post  to  receive  struts. 

Joists. — The  timbers  to  which  the  boards  of  a  floor  or  the  laths  of  a 
ceiling  are  nailed. 

Keep. — The  same  as  donjon,  which  see. 

Key-stone. — The  highest  central  stone  of  an  arch. 

Kiln. — A  building  for  the  accumulation  and  retention  of  heat,  in  or¬ 
der  to  dry  or  burn  certain  materials  deposited  within  it. 

King-post. — The  centre-post  in  a  trussed  roof. 

Knee. — A  convex  bend  in  the  back  of  a  hand-rail.  See  Ramp. 


APPENDIX. 


9 


Ldclarium. — The  same  as  dairy,  which  see. 

Lantern. — A  cupola  having  windows  in  the  sides  for  lighting  an 
apartment  beneath. 

Larmier.— The  same  as  corona,  which  see. 

Lattice. — A  reticulated  window  for  the  admission  of  air,  rather  than 
light,  as  in  dairies  and  cellars. 

Lever-boards. — Blind-slats :  a  set  of  boards  so  fastened  that  they 
may  be  turned  at  any  angle  to  admit  more  or  less  light,  or  to  lap  upon 
each  other  so  as  to  exclude  all  air  or  light  through  apertures. 

Lintel*-*- A  piece  of  timber  or  stone  placed  horizontally  over  a  door, 
■window,  or  other  opening. 

Listel. — The  same  as  fillet,  which  see. 

Lobby. — An  enclosed  space,  or  passage,  communicating  with  the 
principal  room  or  rooms  of  a  house. 

Lodge.— A  small  house  near  and  subordinate  to  the  mansion.  A 
cottage  placed  at  the  gate  of  the  road  leading  to  a  mansion. 

Loop. — A  small  narrow  window.  Loophole  is  a  term  applied  to  the 
Vertical  series  of  doors  in  a  warehouse,  through  which  goods  are  de¬ 
livered  by  means  of  a  crane. 

Luff er -boarding. — The  same  as  lever-boards,  which  see. 

Luthern. — The  same  as  dormer,  which  see. 

Mausoleum-— k  sepulchral  building— so  called  from  a  very  cele¬ 
brated  one  erected  to  the  memory  of  Mausolus,  king  of  Caria,  by  his 
wife  Artemisia. 

Metopa. — The  square  space  in  the  frieze  between  the  triglyphs  of 
the  Doric  order. 

Mezzanine. — A  story  of  small  height  introduced  between  two  of 
greater  height. 

Minaret. — A  slender,  lofty  turret  having  projecting  balconies,  com¬ 
mon  in  Mohammedan  countries. 

Minster. — A  church  to  which  an  ecclesiastical  fraternity  has  been, 
or  is  attached. 

Moat. — An  excavated  reservoir  of  water,  surrounding  a  house,  cas¬ 
tle  or  town. 

Modillion.— A  projection  under  the  corona  of  the  richer  orders,  re¬ 
sembling  a  bracket. 

Module. — The  semi-diameter  of  a  column,  used  by  the  architect  as 
a  measure  by  which  to  proportion  the  parts  of  an  order. 

Monastery. — A  building  or  buildings  appropriated  to  the  reception  of 
monks. 

Monopteron. — A  circular  uollonade  supporting  a  dome  without  an 
•enclosing  wall. 

Mosaic. — A  mode  of  representing  objects  by  the  inlaying  of  small 
•cubes  of  glass,  stone,  marble,  shells,  &c.  . 

Mosque. — A  Mohammedan  temple,  or  place  of  worship. 

Mullions. — The  upright  posts  or  bars,  which  divide  the  lights  in  a 
Gothic  window. 

Muniment-house. — A  strong,  fire-proof  apartment  for  the  keeping 
aud  preservation  of  evidences,  charters,  seals,  &c.,  called  muniments. 

i* 


10 


APPENDIX. 


Museum. — A  repository  of  natural,  scientific  and  literary,  curiosities, 
or  of  works  of  art. 

Mutule. — A  projecting  ornament  of  the  Doric  cornice  supposed  to 
represent  the  ends  of  rafters. 

Nave. — The  main  body  of  a  Gothic  church. 

Newel. — A  post  at  the  starting  or  landing  of  a  flight  of  stairs. 

Niche. — A  cavity  or  hollow  place  in  a  wall  for  the  reception  of  a 
statue,  vase,  &c. 

Nogs. — Wooden  bricks. 

Nosing. — The  rounded  and  projecting  edge  of  a  step  in  stairs. 

Nunnery. — A  building  or  buildings  appropriated  for  the  reception  of 
nuns. 

Obelisk. — A  lofty  pillar  of  a  rectangular  form. 

Octastyle. — A  building  with  eight  columns  in  front. 

Odeum. — Among  the  Greeks,  a  species  of  theatre  wherein  the  poets 
and  musicians  rehearsed  their  compositions  previous  to  the  public  pro¬ 
duction  of  them. 

Ogee. — See  Cyma. 

Orangery. — A  gallery  or  building  in  a  garden  or  parterre  fronting 
the  south. 

Oriel-window. — A  large  bay  or  recessed  window  in  a  hall,  chapel,  or 
other  apartment. 

Ovolo. — A  convex  projecting  moulding  whose  profile  is  the  quad¬ 
rant  of  a  circle. 

Pagoda. — A  temple  or  place  of  worship  in  India. 

Palisade. — A  fence  of  pales  or  stakes  driven  into  the  ground. 

Parapet. — A  small  wall  of  any  material  for  protection  on  the  sides 
of  bridges,  quays,  or  high  buildings. 

Pavilion. — A  turret  or  small  building  generally  insulated  and  com¬ 
prised  under  a  single  roof. 

Pedestal. — A  square  foundation  used  to  elevate  and  sustain  a  co¬ 
lumn,  statue,  &c. 

Pediment. — The  triangular  crowning  part  of  a  portico  or  aperture 
which  terminates  vertically  the  sloping  parts  of  the  roof :  this,  in 
Gothic  architecture,  is  called  a  gable. 

Penitentiary. — A  prison  for  the  confinement  of  criminals  whose 
crimes  are  not  of  a  very  heinous  nature. 

Piazza. — A  square,  open  space  surrounded  by  buildings.  This 
term  is  often  improperly  used  to  denote  a  portico. 

Pier. — A  rectangular  pillar  without  any  regular  base  or  capital. 
The  upright,  narrow  portions  of  walls  between  doors  and  windows  are 
known  by  this  term. 

Pilaster. — A  square  pillar,  sometimes  insulated,  but  more  common 
ly  engaged  in  a  wall,  and  projecting  only  a  part  of  its  thickness. 

Piles. — Large  timbers  driven  into  the  ground  to  make  a  secure 
foundation  in  marshy  places,  or  in  the  bed  of  a  river. 

Pillar. — A  column  of  irregular  form,  always  disengaged,  and  aL 


APPENDIX. 


11 


■ways  deviating  from  the  proportions  of  the  orders  ;  whence  the  distinc¬ 
tion  between  a  pillar  and  a  column. 

Pinnacle. — A  small  spire  used  to  ornament  Gothic  buildings. 

Planceer. — The  same  as  soffit,  which  see. 

Plinth. — The  lower  square  member  of  the  base  of  a  column,  pedes¬ 
tal,  or  wall. 

Porch. — An  exterior  appendage  to  a  building,  forming  a  covered 
approach  to  one  of  its  principal  doorways. 

Portal. — The  arch  over  a  door  or  gate  ;  the  framework  of  the  gate  ; 
the  lesser  gate,  when  there  are  two  of  different  dimensions  at  one  en¬ 
trance. 

Portcullis. — A  strong  timber  gate  to  old  castles,  made  to  slide  up 
and  down  vertically. 

Portico. — A  colonnade  supporting  a  shelter  over  a  walk,  or  ambu¬ 
latory. 

Priory. — A  building  similar  in  its  constitution  to  a  monastery  or 
abbey,  the  head  whereof  was  called  a  prior  or  prioress. 

Prism. — A  solid  bounded  on  the  sides  by  parallelograms,  and  on  the 
ands  by  polygonal  figures  in  parallel  planes. 

Prostyle. — A  building  with  columns  in  front  only. 

Purlines. — Those  pieces  of  timber  which  lie  under  and  at  right  an¬ 
gles  to  the  rafters  to  prevent  them  from  sinking. 

Pycnostyle. — An  intercolumniation  of  one  and  a  half  diameters. 

Pyramid. — A  solid  body  standing  on  a  square,  triangular  or  poly¬ 
gonal  basis,  and  terminating  in  a  point  at  the  top. 

Quarry. — A  place  whence  stones  and  slates  are  procured. 

Quay. — (Pronounced,  key.)  A  bank  formed  towards  the  sea  or  on 
the  side  of  a  river  for  free  passage,  or  for  the  purpose  of  unloading 
merchandise. 

Quoin. — An  external  angle.  See  Rustic  quoins. 

Rabbet ,  or  Rebate. — A  groove  or  channel  in  the  edge  of  a  board. 

Ramp. — A  concave  bend  in  the  back  of  a  hand-rail. 

Rampant  arch. — One  having  abutments  of  different  heights. 

Regula. — The  band  below  the  teenia  in  the  Doric  order. 

Riser. ---In  stairs,  the  vertical  board  forming  the  front  of  a  step. 

Rostrum. — An  elevated  platform  from  which  a  speaker  addresses  an 
audience. 

Rotunda. — A  circular  building. 

Rubble-wall. — A  wall  built  of  unhewn  stone. 

Rudenture. — The  same  as  cable,  which  see. 

Rustic  quoins. — The  stones  placed  on  the  external  angle  of  a  build¬ 
ing,  projecting  beyond  the  face  of  the  wall,  and  having  their  edges 
bevilled. 

Rustic-work. — A  mode  of  building  masonry  wherein  the  faces  of  the 
stones  are  left  rough,  the  sides  only  being  wrought  smooth  where  the 
union  of  the  stones  takes  place. 


12 


APPENDIX. 


Salon,  or  Saloon. — A  lofty  and  spacious  apartment  comprehending' 
the  height  of  two  stories  with  two  tiers  of  windows. 

Sarcophagus. — A  tomb  or  coffin  made  of  one  stone. 

Scantling. — The  measure  to  which  a  piece  of  timber  is  to  be  or  has 
been  cut. 

Scarfing. — The  joining  of  two  pieces  of  timber  by  bolting  or  nailing 
transversely  together,  so  that  the  two  appear  but  one. 

Scotia. — The  hollow  moulding  in  the  base  of  a  column,  between  the 
fillets  of  the  tori. 

Scroll. — A  carved  curvilinear  ornament,  somewhat  resembling  in 
profile  the  turnings  of  a  ram’s  horn. 

Sepulchre. — A  grave,  tomb,  or  place  of  interment. 

Sewer. — A  drain  or  conduit  for  carrying  off  soil  or  water  from  any 
place. 

Shaft. — The  cylindrical  part  between  the  base  and  the  capital  of  a 
column. 

Shoar. — A  piece  of  timber  placed  in  an  oblique  direction  to  support 
a  building  or  wall. 

Sill. — The  horizontal  piece  of  timber  at  the  bottom  of  framing  ;  the 
timber  or  stone  at  the  bottom  of  doors  and  windows. 

Soffit — The  underside  of  an  architrave,  corona,  <5zc.  The  underside 
of  the  heads  of  doors,  windows,  &c. 

Summer. — The  lintel  of  a  door  or  window  ;  a  beam  tenoned  into  a 
girder  to  support  the  ends  of  joists  on  both  sides  of  it. 

Systyle. — An  intercolumniation  of  two  diameters. 

Tcenia. — The  fillet  which  separates  the  Doric  frieze  from  the  archi¬ 
trave. 

Talus. — The  slope  or  inclination  of  a  wall,  among  workmen  called 

Mattering. 

Terrace. — An  area  raised  before  a  building,  above  the  level  of  the 
ground,  to  serve  as  a  walk. 

Tesselated  pavement. — A  curious  pavement  of  Mosaic  work,  com¬ 
posed  of  small  square  stones. 

Tetrastyle. — A  building  having  four  columns  in  front. 

Thatch. — A  covering  of  straw  or  reeds  used  on  the  roofs  of  cottages, 
barns,  &c. 

Theatre. — A  building  appropriated  to  the  representation  of  drama.m 
spectacles. 

Tile. — A  thin  piece  or  plate  of  baked  clay  or  other  material  used  for 
the  external  covering  of  a  roof. 

Tomb. — A  grave,  or  place  for  the  interment  of  a  human  body,  in¬ 
cluding  also  any  commemorative  monument  raised  over  such  a  place. 

Torus. — A  moulding  of  semi-circular  profile  used  in  the  bases  of 
columns. 

Tower. — A  lofty  building  of  several  stories,  round  or  polygonal. 

Transept. — The  transverse  portion  of  a  cruciform  church. 

Transom. — The  beam  across  a  double-lighted  window  ;  if  the  win¬ 
dow  have  no  transom,  it  is  called  a  elere-story  window. 


APPENDIX. 


13 


Tread. — That  part  of  a  stop  which  is  included  between  the  face  of 
its  riser  and  that  of  the  riser  above. 

Trellis. — A  reticulated  framing  made  of  thin  bars  of  wood  for 
screens,  windows,  &c. 

Triglyph. — The  vertical  tablets  in  the  Doric  frieze,  chamfered  on 
the  two  vertical  edges,  and  having  two  channels  in  the  middle. 

Tripod. — A  table  or  seat  with  three  legs. 

Trochilus. — The  same  as  scotra,  which  see. 

Truss. — An  arrangement  of  timbers  for  increasing  the  resistance  to 
cross-strains,  consisting  of  a  tie,  two  struts  and  a  suspending-piece. 

Turret. — A  small  tower,  often  crowning  the  angle  of  a  wall,  &c. 

Tusk — A  short  projection  under  a  tenon  to  increase  its  strength. 

Tympanum. — The  naked  face  of  a  pediment,  included  between  the 
level  and  the  raking  mouldings. 

O  o 

Underpinning. — The  wall  under  the  ground-sills  of  a  building. 

University. — An  assemblage  of  colleges  under  the  supervision  of  a 
senate,  &c. 

Vault. — A  concave  arched  ceiling  resting  upon  two  opposite  paral¬ 
lel  walls. 

Venetian- door. — A  door  having  side-lights. 

Venetian-window. — A  window  having  three  separate  apertures. 

Veranda. — An  awning.  An  open  portico  under  the  extended  roof 
of  a  building. 

Vestibule. — An  apartment  which  serves  as  the  medium  of  commu¬ 
nication  to  another  room  or  series  of  rooms. 

Vestry. — An  apartment  in  a  church,  or  attached  to  it,  for  the  pre¬ 
servation  of  the  sacred  vestments  and  utensils. 

Villa. — A  country-house  for  the  residence  of  an  opulent  person. 

Vinery. — A  house  for  the  cultivation  of  vines. 

Volute. — A  spiral  scroll,  which  forms  the  principal  feature  of  the 
Ionic  and  the  Composite  capitals. 

Voussoirs. — Arch-stones 

Wainscoting. — Wooden  lining  of  walls,  generally  in  panels. 

Water-table. — The  stone  covering  to  the  projecting  foundation  or 
other  walls  of  a  building. 

Well. — The  space  occupied  by  a  flight  of  stairs.  The  space  left 
beyond  the  ends  of  the  steps  is  called  the  well-hole. 

Wicket. — A  small  door  made  in  a  gate. 

Winders. — In  stairs,  steps  not  parallel  to  each  other. 

Zophorus. — The  same  as  frieze ,  which  see. 

Zystos. — Among  the  ancients,  a  portico  of  unusual  length,  common¬ 
ly  appropriated  to  gymnastic  exercises. 


TABLE  OF  SQUARES,  CUBES,  AND  ROOTS. 

(From  Hutton’s  Mathematics.) 


No. 

Square. 

Cube. 

Sq.  Root. 

CubeRoot. 

No. 

Square. 

Cube. 

Sq.  Root. 

CubeRoot. 

1 

1 

1 

1-0000000 

1-000000 

68 

4624 

314432 

8-2462113 

4081655 

2 

4 

8 

1-4142136 

1-259921 

69 

4761 

328509 

8-3066239 

4-101566 

3 

9 

27 

1-7320508 

1-442250 

70 

4900 

343000 

8-3666003 

4-121285 

4 

16 

64 

2-0000000 

1-537401 

71 

5041 

357911 

8-4261498 

4-140818 

5 

25 

125 

2-2360680 

1-709976 

72 

5184 

373248 

8-4852814 

4160168 

6 

36 

216 

2-4494897 

1-817121 

73 

5329 

389017 

85140037 

4  179339 

7 

49 

343 

2-6457513 

1-912931 

74 

5476 

405224 

8-6023253 

4-198336 

8 

64 

512 

2-8284271 

2  000000 

75 

5625 

421875 

8-6602540 

4-217163 

9 

81 

729 

3  0000000 

2-080084 

76 

5776 

433976 

8-7177979 

4-235824 

10 

100 

1000 

3-1622777 

2-154435 

77 

5929 

456533 

8-7749644 

4-254321 

11 

121 

1331 

3-3166248 

2-223980 

78 

6084 

474552 

8-8317609 

4-272659 

12 

144 

1728 

3-4641016 

2-239429 

79 

6211 

493039 

8-8881944 

4-290840 

13 

169 

2197 

3  6955513 

2  351335 

80 

6400 

512000 

8-9442719 

4-303869 

14 

196 

2744 

3-7416574 

2-410142 

81 

6561 

531441 

9-0000000 

4-326749 

15 

225 

3375 

3-8729833 

2-466212 

82 

6724 

551358 

9-0553851 

4-344481 

16 

256 

4096 

4  0000000 

2-519842 

83 

6839 

571787 

9-1104336 

4-362071 

17 

289 

4913 

4-1231056 

2-571232 

84 

7056 

592704 

9  1651514 

4-379519 

18 

324 

5332 

4-2426407 

2-620741 

85 

7225 

614125 

9-2195445 

4-396830 

19 

361 

6859 

4-3538989 

2-663402 

86 

7396 

636056 

9-2736185 

4-414005 

20 

400 

8000 

4-4721350 

2-714418 

87 

7569 

658503 

9-3273791 

4-431048 

21 

441 

9261 

4-5825757 

2-758924 

88 

7744 

681472 

9-3808315 

4-447960 

22 

484 

10643 

4-6904158 

2-802039 

89 

7921 

704969 

9-4339811 

4-464746 

23 

529 

12167 

4-7953315 

2-843367 

90 

8100 

729000 

9-4368330 

4-481405 

24 

576 

13824 

4-8989795 

2-884499 

91 

8281 

753571 

9-5393J20 

4-497941 

25 

625 

15625 

5  0000000 

2-924018 

92 

8164 

778688 

9-5916630 

4-514357 

26 

676 

17576 

5-0990195 

2-962496 

93 

8649 

804357 

9  6436508 

4-530655 

27 

729 

19633 

5  1961524 

3-000000 

94 

8836 

830534 

9-6953597 

4-546836 

28 

784 

21952 

5  2915026 

3036589 

95 

9025 

857375 

9-7467943 

4-562903 

29 

841 

24389 

5-3351643 

3-072317 

96 

9216 

881736 

9-7979590 

4-578857 

30 

900 

27000 

5-4772256 

3-107232 

97 

9409 

912673 

9-8488578 

4-594701 

31 

961 

29791 

5  5677644 

3 141331 

98 

9604 

941192 

9-8994949 

4-610436 

32 

1024 

32768 

5-6568542 

3  174802 

99 

9801 

970299 

9-9498744 

4-626065 

33 

1089 

35937 

5-7445826 

3-207531 

100 

10000 

1000000 

10  0000000 

4-641589 

34 

1156 

39304 

5-8309519 

3  239612 

101 

10201 

1030301 

10-0498755 

4-657009 

35 

1225 

42875 

5  9160798 

3-271066 

102 

10404 

1061208 

100995049 

4-672329 

36 

1296 

46656 

6-0000000 

3  391927 

103 

10609 

1092727 

10-1488916 

4-687548 

37 

1369 

50653 

6-0327625 

3-332222 

104 

10816 

1124861 

10-1980390 

4-702669 

38 

1444 

54872 

6  1644140 

3-361975 

105 

11025 

1157625 

10-2469508 

4-717694 

39 

1521 

59319 

6-2449980 

3-391211 

106 

11236 

1191016 

10-2956301 

4-732623 

40 

1600 

64000 

6-3245553 

3  419952 

107 

11449 

1225043 

10-3140304 

4  747459 

41 

1681 

68921 

6-4031242 

3448217 

108 

11664 

1259712 

10-3923)48 

4762203 

42 

1764 

74088 

6-4807407 

3-476027 

109 

11881 

1295029 

10-4403965 

4-776856 

43 

1849 

79507 

6-5574335 

3-503398 

110 

12100 

1331000 

10-4880885 

4-791420 

44 

1936 

85184 

6  6332496 

3539348 

111 

12321 

1367631 

10-5356538 

4-805895 

45 

2025 

91125 

6-7082039 

3-556893 

112 

12544 

1404928 

10-5330052 

4-820284 

46 

2116 

97336 

6-7823300 

3-533948 

113 

12769 

1442897 

10-6301453 

4-834588 

47 

2209 

103323 

6-8556546 

3-608826 

114 

12996 

1481514 

106770783 

4-848808 

48 

2304 

110592 

6-9232032 

3  634241 

115 

13225 

1520875 

10  7238053 

4-862944 

49 

2401 

117649 

70000000 

3-659396 

116 

13455 

1560896 

10  7703296 

4-876999 

50 

2500 

125000 

7-0710678 

3  634031 

117 

13689 

1601613 

10-8166533 

4-890973 

51 

2601 

132651 

7-1414284 

3-708439 

118 

13924 

1643032 

10-8627805 

4-904868 

52 

2704 

140608 

7-2111026 

3-732511 

119 

14161 

1685159 

10-9087121 

4-918685 

53 

2809 

148877 

7-2301099 

3-756236 

120 

14400 

1728000 

10-9544512 

4-932424 

54 

2916 

157464 

7-3481692 

3-779763 

121 

14641 

1771561 

11-0000000 

4-946087 

55 

3025 

166375 

7-4161985 

3-802952 

122 

14884 

1815848 

11-0453610 

4-959676 

56 

3136 

175616 

7-4833148 

3-825862 

123 

15129 

1860857 

11-0905365 

4-973190 

57 

3249 

185193 

7-5  498344 

3-843501 

124 

15376 

1906624 

11-1355287 

4-986631 

58 

3364 

195112 

7-6157731 

3-870877 

125 

15625 

1953125 

11-1803399 

54)00000 

59 

3481 

205379 

7-6311457 

3-892996 

126 

15876 

2000376 

11-2219722 

5  013298 

6t) 

3600 

216000 

7-7459667 

3-914868 

127 

16129 

2018333 

11-2694277 

5  026526 

61 

3721 

226981 

7-8102497 

3-936497 

128 

16384 

2097152 

11  3137085 

54)39684 

62 

3844 

238328 

7-8740079 

3-957891 

129 

16641 

2146689 

11-3578167 

5  052774 

63 

3969 

250047 

7-9372539 

3-979057 

130 

16900 

2197000 

11-4017543 

5  065797 

64 

4U96 

262144 

8  0000000 

4-000000 

131 

17161 

2248091 

11  4455231 

5078753 

65 

4225 

274625 

8-0622577 

4-020726 

132 

17424 

2299968 

11-4891253 

5091643 

66 

4356 

287496 

8-1240334 

4-041240 

133 

17689 

2352637 

11-5325626 

5  104469 

67 

4489 

300763 

8-1853523 

4-061548 

134 

17956 

2406104 

11-5758369 

5  117230 

APPENDIX 


15 


No. 

Square. 

Cube. 

Sq.  Root. 

CubeRoot. 

No. 

Square. 

Cube. 

Sq.  Root. 

CubeRoot. 

135 

18225 

2460375 

11-6189500 

5-129928 

202 

41  >804 

8242408 

14-2126704 

5-867464 

136 

18496 

2515456 

11-6619033 

5  142563 

203 

41209 

8365427 

14-2478068 

5-877131 

13? 

18769 

2571353 

11-7046999 

5155137 

204 

4 1616 

8483664 

14-2328569 

5-836765 

138 

19044 

2628072 

11-7473401 

5  167649 

205 

4202? 

8615125 

14-3178211 

5-896368 

139 

19321 

2685619 

11-7898261 

5180101 

206 

42 13t 

8741816 

14-3527001 

5-905941 

140 

19600 

2744000 

11-8321596 

5-192494 

207 

4284S 

8869743 

14-3874946 

5915432 

141 

19881 

2803221 

11-8743422 

5-204828 

208 

43264 

8998912 

14-4222051 

5-924992 

142 

20164 

2S63283 

11-9163753 

5-217103 

209 

43681 

9129329 

14-4568323 

5-931473 

143 

20449 

2924207 

11-9532607 

5-229321 

210 

44100 

9261000 

14-4913767 

5-943922 

144 

20736 

2985984 

12-0000000 

5-241483 

211 

44321 

93-03931 

14-5258390 

5-953342 

145 

21025 

3048625 

12-0415946 

5-253588 

212 

44944 

9528123 

14-5602198 

5-962732 

146 

21316 

3112136 

12-0830460 

5-265637 

213 

45369 

9663597 

14-5945195 

5-972093 

147 

21609 

3176523 

12-1243557 

5-277632 

214 

45796 

9800344 

14-6287338 

5-981424 

148 

21904 

3241792 

12-1655251 

5-289572 

215 

46425 

9933375 

14-6623783 

5  990726 

149 

22201 

3307949 

12-2065556 

5  301459 

216 

46656 

10077696 

14-6969385 

6-000000 

150 

22500 

3375000 

12-2474487 

5-313293 

217 

47089 

10218313 

14-7309199 

6  009245 

151 

22301 

3442951 

12-2882057 

5-325074 

218 

47524 

10360232 

14-7618231 

6-018462 

152 

23104 

3511808 

12-3288280 

5-336803 

219 

47961 

10503459 

14-7986186 

6027650 

153 

23409 

3581577 

12-3693169 

5-348481 

220 

48400 

10648000 

14-8323970 

6-036811 

154 

23716 

3652264 

12-4096736 

5-360108 

221 

48341 

10793361 

14-8660687 

6-045943 

155 

24025 

3723375 

12-449899o 

5-371685 

222 

49284 

10941048 

14-8996644 

6-055049 

156 

24336 

3796416 

12-4899960 

5-383213 

223 

49729 

11089567 

14-9331845 

6  064127 

157 

24649 

3869893 

12-5299641 

5-394691 

224 

50176 

1 1239424 

14  9666295 

6073178 

158 

24964 

3944312 

12-5698051 

5-406120 

225 

50625 

1 1390625 

15  0000000 

6-082202 

159 

25281 

4019679 

12-6095202 

5-417501 

226 

51076 

11513176 

15  0332961 

6-091199 

160 

25600 

4096000 

12-6491106 

5-428835 

227 

51529 

1 1697083 

15-0665192 

6100170 

161 

25921 

4173281 

12-6385775 

5-440122 

228 

51984 

1 1852352 

15-0996639 

6-109115 

162 

26244 

4251528 

12-7279221 

5-451362 

229 

52441 

12008939 

15  1327460 

6  118033 

163 

26569 

4330747 

12-7671453 

5-462556 

230 

52900 

12167000 

15  1657509 

6-126926 

164 

26896 

4410944 

12-8062485 

5-473704 

231 

53361 

12326391 

15  1986342 

6  135792 

165 

27225 

4492125 

12-8452326 

5-484807 

232 

53824 

12487168 

15-2315462 

6  144634 

166 

27556 

4574296 

12-8840987 

5  495365 

233 

54239 

12649337 

152643375 

6-153419 

167 

27889 

4657463 

12  9228480 

5-506878 

234 

54756 

12812904 

15  2970585 

6  162210 

168 

28224 

4741632 

12-9614814 

5-517848 

235 

55225 

12977875 

15-3297097 

6171006 

169 

28561 

4826809 

13  0000000 

5-528775 

236 

55696 

13144256 

15  3622915 

6  179747 

170 

28900 

4913000 

130384048 

5-539658 

237 

56169 

13312053 

15-3948043 

6-188463 

171 

29241 

5000211 

13-0766963 

5-550499 

233 

56644 

13481272 

15-4272486 

6-197154 

172 

29584 

5088448 

13  1148770 

5-561298 

239 

57121 

13651919 

15-4596248 

6-205822 

173 

29929 

5177717 

13  1529464 

5-572055 

240 

57600 

13324000 

15-4919334 

6-214465 

174 

30276 

5268024 

13  1909060 

5  532770 

241 

53081 

139*7521 

155241747 

6  223084 

175 

30625 

5359375 

13-2287566 

5-593445 

242 

58564 

14172438 

15-5563192 

6-231630 

176 

30976 

5451776 

13-2664992 

5-604079 

243 

59049 

14348907 

15-5884573 

6-240251 

17? 

31329 

55  45233 

13-3041347 

5-614672 

244 

59536 

14526784 

15-6204994 

6  243800 

178 

31684 

5639752 

13-3416541 

5-625226 

245 

60025 

14706125 

15-6524758 

6-257325 

179 

32041 

5735339 

13-3790882 

5-635741 

246 

60516 

14836936 

15-6843371 

6-265327 

180 

32400 

5832000 

13-4164079 

5  646216 

247 

61009 

15069223 

15-7162336 

6-274305 

181 

32761 

5929741 

13-4536240 

5 -656653 

248 

61504 

15252992 

15-7480157 

6-282761 

182 

33124 

6028568 

13-4907376 

5-667051 

249 

62001 

15433249 

15-7797333 

6  291 195 

183 

33489 

6128487 

13-5277493 

5-677411 

250 

62500 

15625000 

15-8113383 

6-299605 

181 

33856 

6229504 

13-5646600 

5-637734 

251 

63001 

15313251 

15-8429795 

6-397994 

185 

34225 

6331625 

136014705 

5-698019 

252 

63504 

16003008 

15-8745079 

6-316360 

186 

34596 

6434856 

13  6381817 

5-708267 

253 

64009 

16194277 

15-9059737 

6-324704 

187 

34969 

6539203 

13-6747943 

5-718479 

254 

64516 

16337064 

15-9373775 

6-333026 

188 

35344 

6644672 

13-7113092 

5-728654 

255 

65025 

16531375 

15  9637194 

6-341326 

189 

35721 

6751269 

13-7477271 

5-738794 

256 

65536 

16777216 

160000000 

6-349604 

190 

36100 

6859000 

13-7840488 

5-748897 

257 

66049 

16374593 

16  0312195 

6-357861 

191 

36481 

6967871 

13-8202750 

5-758965 

253 

66564 

17173512 

16-0623784 

6-366097 

192 

36864 

7077888 

13-8564065 

5-768998 

259 

67031 

17373979 

16  0934769 

6-374311 

193 

37249 

7189057 

13-8924440 

5-778996 

260 

67600 

17576900 

16  1245155 

6-382504 

194 

37636 

7301334 

13-9283883 

5-788960 

261 

63121 

17779531 

16  1554944 

6-390676 

195 

33025 

7414875 

13-9642400 

5-798890 

262 

6i644 

17984728 

16  1864141 

6-398823 

196 

38416 

7529536 

14  0000000 

5-808786 

263 

69169 

18191447 

16-2172747 

6-406953 

197 

38809 

7645373 

14-0356688 

5-818648 

264 

69696 

18399744 

16-2180763 

6-415069 

198 

39204 

7762392 

14-0712473 

5-823477 

265 

70225 

18609625 

16-2788206 

6-423153 

199 

39601 

7880599 

14  1067360 

5-838272 

266 

70756 

18821096 

16-3095064 

6-431223 

200 

40000 

8000000 

14  1421356 

5-848035 

267 

71289 

19034163 

16-3401346 

6-439277 

201 

40401 

8120601 

14-1774469 

5-837766 

268 

71824 

19248832 

16-3707055 

6-447306 

16 


APPENDIX 


No.  |  Square. 

Cube. 

Sq.  Root.  CubeRoot.j 

No. 

Square. 

Cube. 

Sq  Root. 

^ubeRoot. 

269 

270 

271 

272 

273 

274 

275 

276 

277 

278 

279 

280 
231 
282 
283 

234 

235 
286 
287 
283 
239 

290 

291 

292 

293 

294 

295 

296 

297 

298 

299 

300 

301 

302 

303 

304 

305 

306 

307 

308 

309 

310 

311 

312 

313 
311 
315 
31'i 

317 

318 

319 

320 

321 

322 

323 

324 

325 

326 

327 

328 

329 

330 

331 

332 

333 
331 
335 

72361 
72900 
73441 
73984 
74529 
75076 
75625 
76176 
76729 
77284 
77841 
78400 
78961 
79524 
80089 
80656 
81225 
81796 
82369 
82944 
83521 
84100 
84681 
85264 
85849 
86 136 
87025 
87616 
8 ->209 
88804 
89401 
90000 
90601 
91204 
91809 
92416 
93025 
93636 
94249 
94864 
95481 
96100 
96721 
97344 
97069 
98596 
99225 
99856 
100489 
101124 
101761 
102400 
103041 
103684 
104329 
104976 
105625 
106276 
106929 
107584 
108241 
108900 
109561 
110224 
110889 
111556 
112225 

19465109 
19683000 
19902511 
20123648 
20346417 
20570824 
20796875 
21024576 
21253933 
21484952 
21717639 
21952000 
22188041 
22425768 
22665187 
229063 04 
23149125 
23393656 
23639903 
23887872 
24137569 
24389000 
24642171 
24897083 
25153757 
25412184 
25672375 
25931336 
26198073 
26463592 
26730899 
27000000 
27270901 
27543603 
27818127 
28094464 
28372625 
23652616 
28934443 
29218112 
29503529 
29791000 
30080231 
30371328 
30664297 
30959144 
31255875 
31554496 
31855013 
32157432 
32461759 
32763000 
33076161 
33336248 
33698267 
34012224 
34328125 
34645976 
34965783 
35287552 
35611239 
35937000 
36264691 
36594368 
36926037 
37259704 
37595375 

16-4012195:  6  4553151 

16-4316767!  6-4633041 
16-4620776  6-471274 
16-4924225  6-479224 
16-5227116!  6-487154 
16-5529454  6-495065 
16-5831240  6-502957 
16-6132477  6-510830 
16-6433170  6-518634 
166733320  6  526519 
16-7032931  6-531335 
16-7332005  6-542133 
16-7630546  6-549912 
16-7923556  6-557672 
16-8226038  6-5654 14 
16-8522995  6-573139 
36-8819430  6-530844 

16  9115345  6  538532 
16-9410743  6-596202 

16- 9705627  6-603354 

17  0000000  6-611489 

17- 0293364  6-619106 
17-0537221  6 -626705 1 
17-0880075  6-634237 
17-1172428  6  641852 
17-1464232  6  649400 
17  1755640  6-656930! 
17-2046505  6-664444 
17-2336379  6671940 
17-2626765  6-67942!) 
17-2916165  6-686833 
17  3205081  6694329 
17-3493516  6  701759 
17-3731472  6  709173 
17-4068952  6-716570 
17-4355953  6723951 
17-4642492  6731316 
17-4928557  6-733664 
17-5214155  6  745997 
17-5499288  6753313 
17  5783958  6760614 
17-6068169  6-767899 
17-6351921  6775169 
17-6635217  6  782423 
17-6918060  6  789661 
17-7200451  6796834 
17-7482393  6-804092 
177763383  6-811295 
17-8044933  6-818462 
17-8325545  6-825624 
17-8605711  6-832771 
17-8835433  6-839904 
17-9164729  6-847021 
17-9443584  6  854124 

17- 9722908  6-861212 

18- 0000000  6-868235 
18-0277564  6-875344 
180551701  6-882389 
18-0831413  6-889419 
18-1107703  6-896435 
18-1333571  6  903136 
18-1659021  6-910423 
18-1934054  6-917396 
18-2208672  6-924356 
18-2482376  6-931391 
18-2756669  6-933232 
18-3030052  6-945150 

336 

337 
333 

339 

340 

341 

342 

343 

344 

345 

346 

347 
343 

349 

350 

351 

352 

353 

354 

355 

356 

357 

358 

359 

360 

361 

362 

363 
354 

365 

366 

367 

368 

369 

370 

371 

372 

373 

374 

375 

376 
37: 
3:8 

379 

380 

331 

332 

333 

334 

335 

336 
387 

338 

389 

390 

391 

392 

393 

394 

395 

396 

397 
;  398 
|  399 

400 

401 

!  402 

1 12896 
113569 
114244 
114921 
115600 
116281 
116964 
117649 
118336 
119025 
119716 
120409 
121104 
121801 
122500 
123201 
123904 
124609 
125316 
126025 
126736 
127449 
128164 
128881 
129600 
130321 
131044 
131769 
132496 
133225 
133956 
131689 
135124 
136161 
136900 
137641 
138384 
139129 
139876 
140625 
111376 
142129 
142884 
143641 
144400 
' 145161 
145924 
146689 
147456 
148225 
148996 
149769 
150544 
151321 
152100 
152831 
153664 
154449 
155236 
156025 
155816 
157609 
158404 
159201 
160000 
160801 
161604 

37933056 

38272753 

33614472 

38958219 

39304000 

39651821 

40001688 

40353607 

40707584 

41063625 

41421736 

41781923 

42144192 

42508549 

42875000 

43243551 

43614208 

4398  >977 

44361864 

44733875 

45118016 

45499293 

45382712 

46268279 

46656000 

47015381 

47437928 

47832147 

48228544 

48627125 

49027393 

49430863 

49836032 

50243409 

50653000 

51064811 

51478848 

51895117 

52313624 

52734375 

53157376 

53582633 

54010152 

54439939 

54872000 

55306341 

55742968 

56181887 

56623104 

57066625 

57512456 

57960603 

58411072 

58863869 

59319000 

59776471 

60236238 

60693457 

61162984 

61629875 

62099136 

62570773 

63044792 

63521199 

64000000 

64481201 

64964808 

18-3303028 
18  3575598 
18-3347763 
18-4119526 
18-4390889 
18  4661853 
18-4932420 
18-5202592 
18-5472370 
18-5741756 
18-6010752 
18-6279360 
18-6547581 
18-6815417 
18-7082869 
18-7349940 

18  7616630 
18-7832942 
188148377 
18-8414437 
18-8679623 
18-8944436 
18-9208379 
18-9472953 

18- 9736660 

19- 0000000 
19-0262976 
19-0525589 

19  0787840 
19  1049732 
191311265 
19-1572441 
19-1833261 
19  2093727 
19-2353341 
19-2613603 
19-2373015 
19-3132079 
19-3390796 
19-3649167 
19-3907194 
19-4164878 
19-4422221 
19-4679223 
19-4935387 
195192213 
19-5448203 
19-5703358 
19-5959179 
19-6214169 
19-6468327 

19  6723156 
,  19-6977156 

19-7230829 

19-7484177 

19-7737199 

19-7989899 

19-8242276 

19-8494332 

19-8746069 

19-8997487 

19-9248538 

19-9499373 

19- 9749844 

20- 0000000 

20  0249844 
I  20  0499377 

6  952053 
6-953913 
6-965820 
6-972683 
6-979532 
6-986368 

6- 993191 

7  000000 

7- 006796 
7-013579 

7  020349 

7  027106 
7-033350 

7  040581 
7-047299 
.7  051004 

7  060697 
7067377 

7  074044 
7-080699 
7-087341 
7093971 

7  100538 
7107194 

7  113787 

7  120367 

7  126936 

7  133492 

7  140037 

7  146569 
7 153090 
7-159599 
7-166096 
7-172591 
7-179054 
7  185516 
7191966 
7-198405 
7-204832 
7-211248 
7-217652 
7-224045 
7-230427 
7-236797 
7-243156 
7-249504 
7-255341 
7-262167 
7-263482 
7-274786 
7-231079 
7-287362 
7-293633 
7-299894 
7-306144 
7  312383 
7318611 
7-324829 
7  331037 
7-337234 
7-343420 
7349597 
7-355762 
7-361918 
7  368063 
7  374198 
7-330323 

Appendix* 


17 


No. 

Square. 

Cube. 

Sq.  Root. 

CubeRoot 

No. 

Square. 

Cube. 

Sq.  Root. 

CubeRoot. 

403 

404 

405 

406 
40? 

408 

409 

410 

411 

412 

413 
4 1 1 
415 
41t- 

417 

418 

419 

420 

421 

422 

423 

424 

425 

426 

427 

428 

429 

430 

431 

432 

433 

434 

435 

436 

437 
433 

439 

440 

441 

442 

443 

444 

445 

446 

447 

448 

449 

450 

451 

452 

453 

454 

455 

456 

457 

458 

459 

460 

461 

462 

463 

464 

465 

466 

467 

468 

469 

162409 
163216 
164025 
164836 
165649 
166464 
167281 
168100 
168921 
169744 
170569 
171396 
172225 
173056 
173889 
174724 
175561 
176400 
177241 
178084 
178929 
179776 
180625 
181476 
182329 
183184 
1840 tl 
184900 
185761 
186624 
187489 
188356 
189225 
190096 
190969 
191844 
192721 
193600 
194481 
195364 
196219 
197136 
198025 
198916 
199809 
200704 
201601 
202500 
203401 
204304 
205209 
206116 
207025 
207936 
208849 
209764 
210681 
211600 
212521 
213444 
214369 
215296 
216225 
217156 
218089 
219024 
219961 

65450827 
65939264 
66430125 
66923416 
67419143 
67917312 
68417929 
68921000 
69426531 
69934528 
70444997 
70957944 
71473375 
71991296 
72511713 
73034632 
73560059 
74088000 
74618461 
75151448 
75686967 
76225024 
76765625 
77308776 
77854483 
78402752 
78953589 
79507000 
80062991 
80621568 
81182737 
81746504 
82312875 
82881856 
83453453 
84027672 
84604519 
85184000 
8576  .121 
86350888 
86938307 
87528384 
88121125 
88716536 
89314623 
89915392 
90518849 
91125000 
91733851 
92345408 
92959677 
93576664 
94196375 
94818816 
95443993 
96071912 
96702579 
97336000 
97972181 
98511128 
99252847 
99897344 
100544625 
101194696 
101847563 
102503232 
103161709 

20  0748599 
20  0997512 
20  1246118 
20  1494417 

20  1742410 
20-1990099 
20-2237484 
20-2484567 
20-2731349 
20-2977831 
20-3224014 
20-3469899 
20-3715488 
20-3960781 
20-4205779 
20-4450483 
20-4694895 
20-4939015 
20-5182845 
20-5426386 
20-5669638 
20-5912603 
20-6155281 
20-6397674 
20-6639783 
20-6881609 
20-7123152 
20-7364414 
20-7605395 
20-7846097 
20-8086520 
20-8326667 
20-8566533 
20-8806130 
20-9045450 
20-9284495 
20-9523268 

20- 9761770 

21- 0000000 
21-0237960 
21-0475652 
21-0713J75 

21  0950231 
21-1187121 
21-1423745 
21-1660105 
21-1896201 
21-2132034 
21-2367606 
21-2602916 
21-2837967 
21-3072758 
21-3307290 
21-3541565 
21-3775583 
21-4009346 
21-4242853 
21-4476106 
21-4709106 
21-4941853 
21-5174318 
21-5406592 
21-5638587 
21-5870331 
21-6101828 
21-6333077 
21-6564078 

7  386437 
7-392542 
7398636 
7-404721 
7-410795 
7-416859 
7-422914 
7-428959 
7-434994 
7-441019 
7-447034 
7-453040 
7-459035 
7-465022 
7-470999 
7-476966 
7-482924 
7-488872 
7-494811 
7-500741 
7-506661 
7-512571 
7-518473 
7-524365 
7-530248 
7-536122 
7-541987 
7-547842 
7-553689 
7-559526 
7  565355 
7-571174 
7-576985 
7-582786 
7-588579 
7-504363 
7600138 
7-605905 
7-611663 
7-617412 
7-623152 
7-628881 
7-634607 
7  640321 
7-646027 
7-651725 
7-657414 
7-663094 
7-668766 
7674430 
7-680086 
7-685733 
7691372 
7  697002 
7-702625 
7-708239 
7713845 
7-719443 
7-725032 
7-730614 
7-736188 
7-741753 
7-747311 
7-752861 
7-758402 
7-763J36 
7-769462 

470 

471 

472 

473 

474 

475 

476 

477 

478 

479 

480 

481 

432 

433 

434 

435 
486 
437 
488 
439 

490 

491 

492 

493 

494 

495 

496 

497 

498 

499 

500 

501 

502 

503 

504 

505 

506 

507 

508 

509 

510 

511 

512 

513 

514 

515 

516 

517 

518 

519 

520 

521 

522 

523 

524 

525 

526 

527 

528 

529 

530 

531 

532 

533 

534 

535 
|  536 

220900 

221841 

222784 

223729 

224676 

225625 

226576 

227529 

228484 

229441 

230400 

231361 

232324 

233289 

234256 

235225 

236196 

237169 

238144 

239121 

240100 

241081 

212064 

243049 

244036 

245025 

246016 

247009 

248004 

249001 

250000 

2510Q1 

252004 

253009 

254016 

255025 

256036 

257049 

258064 

259081 

260100 

261121 

262144 

263169 

264196 

265225 

266256 

267289 

268324 

269361 

270400 

271441 

272434 

273529 

274576 

275625 

276676 

277729 

278734 

279341 

280900 

281961 

283024 

284089 

285156 

286225 

287296 

103823000 

104487111 

105154048 

105823817 

106496424 

107171875 

107850176 

108531333 

109215352 

109902239 

110592000 

111284641 

111930168 

112678587 

113379904 

114084125 

114791256 

115501303 

116214272 

116930169 

117649000 

118370771 

119095488 

119823157 

120553784 

121287375 

122023936 

122763473 

123505992 

124251499 

125000000 

125751501 

126506008 

127263527 

128024064 

12878762. 

129554216 

13032.^843 

131090512 

131872229 

132651000 

133432031 

134217728 

135005697 

135796744 

136540875 

137388096 

138180413 

138991832 

139798359 

140608000 

141420761 

142236648 

143055667 

143877824 

144703125 

145531576 

146363183 

147197952 

148035889 

148877000 

149721291 

150568768 

151419437 

152273304 

153130375 

153990656 

21-6794334 

21-7025344 

21-7255610 

21-7485632 

21-7715411 

21-7944947 

21-8174242 

21-840329? 

21-8632111 

21-8860686 

21-9089023 

21-9317122 

21-9544984 

21- 9772610 

22- 0000000 
220227155 
22  0451077 
22-0680765 
220907220 
22  1133444 
22  1359436 
22-1585198 
22-1810730 
22  2036033 
22  2261108 
222485955 
222710575 
222934963 
22-3159136 
22-3383079 
223606798 

22- 3830293 
22  4053565 

23- 4276615 
22  4499443 
22-472^051 
22-4944438 
225166605 
22  5388553 

22  5610383 
22-5831796 
22-6O53091 
22-6274170- 
22-6*95033 
22-6715681 
22-6936114 
227156334 
22-7376340 
22-7596134 
22-7815715 
22-8035085 
22-8354344 
22-8473193 
22-8691933 
22-8910463 
22-9128785 
22-9346 -.99 
22-9554806 

22- 9782506 

23  0000000 
23  0217289 

23- 0434372 
23-0651252 
23  0867928 
23  1084400 
2313.H8670 
23  1516738 

7-774980 
7-780490 
7-785993 
7-791487 
7-796974 
7-802454 
7-807925 
7-813389 
7-818846 
7-824294 
7-829735 
7-835169 
7-810595 
7-816013 
7-851424 
7-856828 
7-862224 
7-867613 
7-872994 
7-878368 
7-883735 
7-839095 
7-894447 
7-899792 
7-905129 
7-910460 
7-915783 
7-921099 
7-926408 
7-931710 
7-9370U5 
7-942293 
7-947574 
7-952848 
7-953114 
7-963374 
7-96862? 
7-973873 
7-979 11a 
7-984314 
7-989570 

7- 994788 
8  000000 

8- 005205 
8-01..403 
8-015595 
8-020779 
8-025957 
8-031129 
8-036293 
8-041451 
8-046603 
8-051748 
8  056886 
8-062018 
8-067143 
8-072262 
8-077374 
8-082480 
8-087579 
809267a 
8-097759 
8-102839 
8-107913 
8-112980 
8-118041 
8-123091. 

3* 


18 


APPENDIX 


No. 

Square. 

Cube. 

Sq.  Root. 

CubeRoot- 

No. 

Square. 

Cube. 

Sq.  Root. 

CubeRoot. 

537 

533 

539 

540 

541 

542 

543 

544 

545 

546 

547 

548 

549 

550 

551 

552 

553 

554 

555 

556 

557 

558 

559 

560 

561 

562 

563 

564 

565 

566 

567 

568 

569 

570 

571 

572 

573 

574 

575 

576 

577 

578 

579 

580 

581 

582 

583 

534 

535 

586 

587 

588 

589 

590 

591 
5'92 

593 

594 

595 

596 

597 

598 

599 

600 
601 
602 
603 

288369 

239444 

290521 

291600 

292681 

293764 

294849 

295936 

297025 

298116 

299209 

300304 

301401 

302500 

303601 

304704 

305809 

306916 

308025 

309136 

310249 

311364 

312481 

313600 

314721 

315344 

316969 

318096 

319225 

320356 

321489 

322624 

323761 

324900 

326041 

327184 

328329 

329476 

330625 

331776 

332929 

334034 

335241 

336400 

337561 

333724 

339889 

341056 

342225 

343396 

344569 

345744 

346921 

348100 

349281 

350464 

361649 

352836 

354025 

355216 

356409 

357604 

358801 

360000 

361201 

362404 

363609 

154854153 

155720872 

156590819 

157464000 

158340421 

159220088 

160103007 

160989184 

161878625 

162771336 

163667323 

164566592 

165469149 

166375000 

167284151 

168196608 

169112377 

170031464 

170953875 

171879616 

172808693 

173741112 

174676879 

175616000 

176558481 

177504328 

178453547 

179406144 

180362125 

181321496 

182284263 

183250432 

184220009 

185193000 

186169411 

187149248 

189132517 

189119224 

190109375 

191102976 

192100033 

193100552 

194104539 

195112000 

196122941 

197137368 

198155287 

199176704 

200201625 

201230056 

202262003 

203297472 

204336469 

205379000 

206425071 

207474688 

208527857 

209584584 

210644875 

211708736 

212776173 

213947192 

214921799 

216000000 

217081801 

218167208 

219256227 

23  1732605 
23- 1948270 
23-2163735 
23-2379001 
23-2594067. 
23  2908935 
23-3023604 
23-3239076 
23-3452351 
23-3666429 
23-3880311 
23-4093998 
23-4307490 
23-4520788 
23-4733392 
23-4946802 
23-5159520 
235372046 
23  5534380 
23-5796522 
23-6008474 

23  6220236 
23-6431808 
23-6643191 
23-6854386 
23-7065392 
23-7276210 
23-7486842 
23-7697286 
23-7907545 
23-8117618 
23-8327506 
23-8537209 
23-8746728 
238956063 
23-9165215 
23-9374184 
23-9582971 

23- 9791576 

24- 0000000 
24-0208243 
24-0416306 
24-0624188 
24-0831891 
24-1039416 
24-1246762 
24-1453929 
24-1660919 
24-1867732 
24-2074369 
24-2230829 

24  2487113 
24-2693222 
24-2899156 
24-3104916 
24  3310501 
24-3515913 
24-3721152 
24-3926218 
24-4131112 
24-4335834 
24-4540385 
24.4744765 
24  4948974 
24-5153013 
24-5356383 
24-5560583 

8123145 
8-133187 
8-133223 
8-143253 
8-148276 
8-153294 
8-158305 
8-163310 
8-168309 
8-173302 
8-178289 
8-183269 
8-188244 
8  193213 
8198175 
8-203132 
8-203082 
8-213027 
8-217966 
8-222893 
8-227825 
8-232746 
8-237661 
8-242571 
8-247474 
8-252371 
8-257263 
8-262149 
8-267029 
8-271904 
8-276773 
8-281635 
8-286493 
8-291344 
8-296190 
8-301030 
8-305865 
8-310694 
8-315517 
8-3203  <5 
8-325147 
8-329954 
8-334755 
8-339551 
8-344341 
8-349126 
8-353905 
8-358678 
8-363447 
8-368209 
8-372967 
8-377719 
8-382465 
8-387206 
8-391942 
8-396673 
8-401398 
8-406118 
8-410333 
8-415542 
8-420246 
8-424945 
8-429633 
8-434327 
8-439010 
8-443688 
8-448360 

604 

605 

606 

607 

608 

609 

610 
611 
612 

613 

614 

615 

616 

617 

618 

619 

620 
621 
622 

623 

624 

625 

626 

627 

628 

629 

630 

631 

632 

633 

634 

635 

636 

637 

638 

639 

640 

641 

642 

643 
641 

645 

646 

647 

648 

649 

650 

651 

652 

653 

654 

655 

656 

657 

658 

659 

660 
661 
662 

663 

664 

665 

666 

667 

668 
j  669 
1  670 

304816 
306025 
367236 
368449 
369664 
370881 
372100 
373321 
374554 
375769 
376996 
378225 
379456 
380689 
381924 
383161 
384400 
385641 
386884 
38tl29 
389376 
390625 
391876 
393129 
394384 
395641 
396900 
388161 
389424 
400689 
401956 
403225 
404(96 
405769 
407044 
408321 
4096O0 
410881 
412164 
413449 
414736 
416025 
417316 
418609 
419904 
421201 
422500 
423801 
425104 
426409 
427716 
429025 
43  336 
431649 
432964 
434281 
435600 
436921 
438244 
439569 
440896 
442225 
443556 
444889 
446224 
447561 
448900 

220348864 

221445125 

222545016 

223648543 

224755712 

225866529 

226981000 

228099131 

229220928 

230346397 

231475544 

232608375 

233744896 

234885113 

236029032 

237176659 

238328000 

239483061 

240641848 

241804367 

242970624 

244140625 

245314376 

246491883 

247673152 

248858189 

250047000 

251239591 

252435968 

253636137 

254840104 

256047875 

257259456 

258474853 

259694072 

260917119 

262144000 

263374721 

264609288 

265847707 

267089984 

268336125 

269586136 

270840023 

272097792 

273359449 

274625000 

275894451 

277167808 

278445077 

279726264 

281011375 

282300416 

283593393 

284890312 

236191179 

287496000 

288804781 

290117528 

291434247 

292754944 

294079625 

295408296 

296740963 

298077632 

299418309 

300763000 

24-5764115 

24-5967478 

24-6170673 

24-6373700 

24-6576560 

24-6779254 

24  6981781 
24-7184142 
24-7386338 
24-7588368 
24-7790234 
24-7991995 
24-8193473 
24-8394847 
24-8596058 
24-8797106 
24-8997992 
24-9198716 
24-9399278 
24-9599679 

24- 9799920 

25  0000000 
25  0199920 

25- 0399681 
25  0599282 
25-0798724 
25  0998008 
25  1197134 
25  1396102 
25  1594913 
25  1793566 
25-1992063 
252190404 
25  2338589 
25-2586619 
25  2784493 
25.2982213 
25-3179778 
25-3377189 
25-3574447 
25  3771551 
253968502 
25  4165301 
25-4361947 
25  4558441 
25  4754784 
25  4950976 
25  5147016 
25  5342907 
25  5538647 
25-5734237 
255929678 
25  6124969 
25  6320112 
256515107 
25  6709953 
25  6904652 
25-7099203 
25  7293607 
25  7487864 
25  7681975 
25  7875939 
25  8069758 
25  8263431 
25-8456960 
25-8650343 
25-8843582 

8-453028 
8-457691 
8-462348 
8-467000 
8-471647 
8-476289 
8-480926 
8-485558 
8490185 
8-494806 
8-499423 
8-504035 
8-508642 
8-513243 
8-517840 
8-522432 
8-527019 
8-531601 
8-536178 
8-540750 
8-545317 
8-549880 
8-554437 
8-558990 
8-563538 
8-568081 
8-572619 
8-577152 
8-581681 
8-586205 
8  590724 
8-595238 
8-599748 
8604252 
8-608753 
8-613248 
8-617739 
8-622225 
8-626706 
8-631183 
8-635655 
8.640123 
8-644585 
8-649044 
8-653497 
8-657946 
8-662391 
8-666831 
8-671266 
8-675697 
8-680124 
8-684546 
8-688963 
8693376 
8697784 
8-702188 
8-706538 
8-710983 
8-715373 
8-719760 
8  724141 
8-728518 
8  732892 
8-737260 
8741625 
8-745985 
8-750340 

APPENDIX, 


19 


No. 

Square. 

Cube. 

Sq.  Root. 

CubeRoot. 

No. 

Square. 

Cube. 

Sq.  Root. 

CubeRoot. 

671 

671 

673 

674 

675 

676 

677 

678 

679 

680 
681 
682 

683 

684 

685 

686 

687 

688 

689 

690 

691 

692 

693 
o94 

695 

696 

697 

698 

699 

700 
701 
702 

703 

704 

705 

706 

707 

708 

709 

710 

711 

712 

713 

714 

715 

716 

717 

718 

719 

720 

721 

722 

723 

724 

725 

726 

727 

728 

729 

730 

731 

732 

733 

734 

735 

736 

737 

450241 

451584 

452929 

454276 

455625 

456976 

458329 

459684 

461041 

462400 

463761 

465124 

466489 

467856 

469225 

470596 

471969 

473344 

474721 

476100 

477481 

478864 

480249 

481636 

483025 

484416 

485809 

487204 

488601 

4900 00 

491401 

492-504 

494209 

495616 

497025 

498436 

499849 

501264 

502681 

504100 

505521 

506944 

508369 

500796 

511225 

512656 

514089 

515524 

516961 

518400 

519841 

521284 

522729 

524176 

525625 

527076 

528529 

529984 

531441 

532900 

534361 

535824 

527289 

53o756 

540225 

54l69o 

543169 

302111711 

303464448 

304821217 

306182024 

307546875 

308915776 

310288733 

311665752 

313046839 

314432000 

315821241 

317214568 

318611987 

320013504 

321419125 

322828850 

324242703 

325660672 

327082769' 

328509000 

329939371 

331373888 

332812557 

334255384 

335702375 

337153530 

338608873 

340068392 

341532099 

343000000 

344472101 

345948408 

347428927 

3-18913661 

35040^625 

351895816 

353393243 

354894912 

356400829 

357911000 

359425431 

360944128 

362467097 

303994344 

365525875 

367061,696 

368601813 

370146232 

371694959 

373248000 

3748U5361 

370367048 

377933067 

379503424 

381078125 

382657176 

384240583 

385828352 

.387420489 

389017000 

390617891 

392223168 

393832837 

3J5446904 

397065375 

398688256 

400315553 

25-9036677 

25-9229628 

25-9422435 

25-9615100 

25- 9807621 
26  0000000 

26- 0192237 
26-0384331 
26.0576284 
26-0768096 
26  0959767 
26-1151297 
26-1342687 
26-1533937 
26-1725047 
26-1916017 
26-2106848 
26-2297541 
26-2483095 
26-2678511 
26-2868789 
26-3053929 
26-3248932 
26-34387^7 
26-3628527 
26-3318119 
26-4007576 
26-4196896 
26-4386081 
26-4575131 
26-4764046 
26-4952826 
26  5141472 
26  5329983 
26-5518361 
26-5706605 
26-5894716 
26  6082694 
26-6270539 
26-6458252 

26  6645833 
26-6833281 
26-7020598 
26-7207784 
26-7394839 
26-7581763 
26-7768557 
26-7955220 
26-8141754 
26-8328157 
26-8514432 
268700577 
268886593 
26-9072481 
26-9258240 
26-9443872 
26-9629375 

26- 9814751 

27  0000000 

27- 0185122 
27  0370117 
27-0554985 
27-0739727 
27-0924344 
27.1108834 
27-1293199 
27-1477439 

8-754691 

8-759033 

8-763331 

8-767719 

8-772053 

8-776333 

8-780708 

8-785030 

8-789347 

8-793659 

8-797968 

8-802272 

8-806572 

8-810868 

8-815160 

8-819447 

8-823731 

8-828010 

8-832285 

8-836556 

8-840823 

8-845085 

8-84^344 

8-853598 

8-857849 

8-862095 

8.866337 

8-870576 

8-874810 

8-879040 

8-883266 

8-887483 

8-891706 

8-895920 

8900130 

8-904337 

8-908539 

8-912737 

8-916931 

8-921121 

8-925308 

8-929490 

8-933669 

8-937843 

8-942014 

8-946181 

8-950344 

8-954503 

8-958658 

8-962809 

8-966957 

8-971101 

8-975241 

8-979377 

8-983509 

8-987637 

8-991762 

8- 995883 

9- 000000 
9-004113 
9-008223 
9012329 
9-016431 
9  020529 
9-024624 
9-023715 
9-032802 

738 

739 

740 

741 

742 

743 

744 

745 

746 

747 

748 

749 

750 

751 

752 

753 

754 

755 

756 

757 

758 

759 

760 

761 

762 

763 

764 

765 

766 

767 

768 

769 

770 

771 

772 

773 

774 

775 

776 

777 

778 

779 

786 

781 

782 

783 

784 

785 

786 
78/ 

788 

789 

790 

791 

792 

793 
79* 

795 

796 
79/ 

798 

799 

800 
801 
802 

803 

804 

544644 

546121 

547600 

549081 

550564 

552049 

553536 

555025 

556516 

558009 

559504 

561001 

562500 

564o01 

565504 

567009 

568516 

570025 

571536 

573649 

574564 

576081 

577600 

579121 

580644 

582169 

583696 

585225 

586756 

588289 

589824 

591361 

592900 

594441 

5n5a84 

597529 

599076 

600625 

602176 

603729 

605284 

606841 

6084x0 

60j961 

611524 

613089 

614656 

616225 

617796 

619369 

620944 

622521 

624100 

625681 

627264 

628849 

630436 

632025 

63361b 

635209 

636804 

638401 

640000 

641601 

643204 

644809 

646416 

401947272 

403583419 

405224000 

406869021 

408518488 

410172407 

411830784 

413493625 

415160936 

416832723 

418508992 

420189749 

421875000 

423564751 

425259008 

426957777 

423661064 

430368875 

432081216 

433798093 

435519512 

437245479 

438976000 

440711081 

442450728 

444194947 

445943744 

447697125 

449455096 

451217663 

452984832 

454756609 

45b533000 

458314011 

400099648 

461889917 

463684824 

465484375 

4b7288576 

469097433 

470910952 

472729139 

474552000 

476379541 

47o211768 

480048687 

481890304 

483736625 

4855a7b5b 

487443403 

489303872 

491169069 

493039000 

494913671 

496793080 

4986/7257 

500566184 

502459875 

504358336 

506261573 

508169592 

510082399 

512000000 

513922401 

515849608 

517781627 

519718464 

27  1661554 
27-1845544 
27-2029410 
27-2213152 
27-2396769 
27-2580263 
27-2763634 
27-2946881 
27-3130006 
27-3313007 
27-3495887 
27-3678644 
27-3061279 
27-4043792 
27-4226184 
27-4408455 
27-4590604 
27-4772633 
27-4954542 
27-513b33x 
27-5317998 
27-5499546 
27-5680975 
27-58b2284 
27-6043475 
27  6224546 
27'64u5499 
27  658o334 
27  b7o7o50 
27-6947640 
27-7128129 
27-7308492 
27-7488739 
27-7668860 
27-7o4)000 
278028775 
27  8208555 
27-0388218 

27  856776b 
27-874719.7 
27-8926514 
27-9105715 
27-9284801 
27-9463772 
27-9642629 

27- 9821372 

28- 0000000 

28  Ol7o5l5 
280356915 
28  0535203 
28  0713377 
28  0891430 
28- 106938b 
28-1247222 
28- 142494b 
28-1602557 
28  1780056 
28-1957444 
28-213472U 
28-2311884 
28-2488930 
28-2665881 
28-2842712 
28-3019434 
28-319x045 
28337254b 
28-3548938 

9  036886 
9040965 
9045042 
9049114 
9053183 
9-057248 
9061310 
9065368 
9-069422 
9073473 
907/520 

9  Os 1563 
9-085b03 
9-089639 
9-0936/2 
9-097701 
9-101726 
9-lu574b 
9-109767 
9-113702 
9-117793 
9-121801 
9-125805 
9-1298x6 
9-133803 
9-137797 
9-141787 
9-14577* 

9  14975s 
9-153737 
9-15/7i4 
9- lb 1687 
9-165656 
9-169022 
9- 1/358 j 
9-177544 
9181500 
9-185453 
9-189402 

9  193347 
9-197290 
9-201229 
9  205104 
9-203096 
9-213025 
9-216950 
9-220873 
9-224791 
9-22o7o7 
9-202619 
y-2oo528 
9-240435 
9-244333 
9-248234 
9-252130 
9-250022 
9-209911 
9-26.*797 
9-2b7bbo 
9-271559 
9-275435 
9-279308 
9-283176 
9-287044 
9-290a07 
9-29476/ 
9-298624 

20 


APPENDIX 


No. 

805 

806 

807 

808 

809 

810 
811 
812 

813 

814 

815 

816 

817 

818 

819 

820 
821 
822 

823 

824 

825 

826 

827 

828 

829 

830 

831 

832 

833 

834 

835 

836 

837 

838 

839 

840 

841 

842 

843 

844 

845 

846 

847 

848 

849 

850 

851 

852 

853 

854 

855 

856 

857 

858 

859 
86C 
861 
862 

863 

864 

865 

866 

867 

868 
869 
87( 
87] 

Square. 

Cube. 

Sq.  Root,  t 

^ubeRootJ 

No. 

Square. 

Cube. 

Sq.  Root.  C 

lubeRoot, 

648025 
649636 
651249 
652864 
654181 
656100 
657721 
659344 
660969 
662596 
664225 
665856 
667489 
669124 
670761 
672400 
674041 
675684 
677329 
678976 
680625 
682276 
633929 
685584 
687241 
6889  iO 
690561 
69*224 
693889 
695556 
697225 
698896 
700569 
702244 
703921 
705600 
707281 
708964 
710649 
712336 
714025 
715716 
717409 
719104 
720801 
722506 
724201 
725904 
72760, 
729316 
7310*f 
732736 
734441 
736164 
73788] 
739606 
741321 
74304, 
744761 
746491 
748*2f 
749951 
75168' 
75342' 
75516 
756901 
1  75864 

521660125! 
52360G616 
525557943 
527514112 
529475129 
531441000 
533411731 
535387328 
537367797 
539353144 
541343375 
543338496 
545338513 
547343432 
549353259 
551368000 
553387661 
555412248 
557441767 
559476224 
561515625 
563559976 
565609283 
567663552 
569722789 
571787000 
573856191 
57593j368 
578009537 
580093704 
582182875 
584277056 
586376253 
588480472 
590589719 
592704000 
594823321 
596947688 
599077107 
601211584 
6o335 1 125 
605495736 
607645423 
609800192 
611960049 
614125000 
6162^5051 
618470208 
620650477 
622835864 
625026375 
627222016 
629422793 
631628712 
633839779 
636056000 
638277381 
640503928 
642735647 
644972544 
647214625 
5  64946 J89fc 
651714363 
1  653972032 
656234909 
J  65S50300C 
66077631] 

28  3725219 
28-3901391 
28-4077454 
28  4253408 
28-4429253 
28-4604989 
28-4780617 
28-4956137 
28-5131549 
28-5306852 
28-5482048 
28-5657137 
28-5832119 
28-6006993 
28-6181760 
28-6356421 
28-6530976 
28-6705424 
23-6879766 
28-7054002 
28-7228132 
28-7402157 
28-7576077 
28-7749891 
28-7923601 
28-8097206 
28-8270706 
28-8444102 
28-8617394 
28-8790582 
28-8963666 
28-9136646 
28-9309523 
28-9432297 
28-9654967 

28- 9827535 
29  0000000 
290172363 
29  0344623 

29- 0516781 
29-0688837 
29-086079 1 
29-1032644 
29-1204396 
29  1376046 
29  1547595 
29  1719043 
29-1890390 
29-2061637 
29-2232784 
29-2403830 
29-2574777 
29-2745623 
29  2916371 
29-3087018 
29-3257561 
29-34230  IS 
29-3598363 
29-3768610 
29-3933761 
29-4108823 
29-427877* 
29-444863' 
29-461839' 
29-4788051 
29-4957621 
29-512709] 

9-302477 

9-306328 

9-310175 

9-314019 

9-317860 

9-321697 

9-325532 

9-320363 

9-333192 

9-337017 

9340839: 

9-344657 

9-348473 

9-352286 

9-356095 

9-359902 

9  363705 

9  367505’ 

9-371302! 

9-375096 

9-378887 

9-382675 

9-386460 

9-390242 

9-394021 

9-3,7796 

9-401569! 

9-405339 

9-409105 

9-412869 

9-416630 

9-420387 

9-424142 

9-427844 

9-431642 

9-435388 

9-439131 

9-442870 

9-446607 

9-450341 

9-454072 

9-457800 

9-461525 

9-465247 

9-468966 

9-472682 

9-476396 

9-4801U6 

9-483814 

9-487518 

9-491220 

9-494919 

9-498615 

9-502308 

9-505998 

9-509685 

9-513370 

9517051 

9-520730 

9524406 

9-528079 

9-531750 

9-535417 

9-539082 

9-542744 

9546403 

9-55005S 

872 

873 

874 

875 

876 

877 

878 

879 

880 
881 
882 

883 

884 

885 

886 

887 

888 

889 

890 

891 

892 

893 

894 

895 

896 

897 

898 

899 

900 

901 

902 

903 

904 

905 

906 

907 

908 

909 

910 

911 

912 

913 
91 1 

915 

916 
911 

918 

919 
92, 

921 

922 

923 
!  924 
!  925 
l  926 

927 
926 
92* 
i  931 
!  93] 
|  932 
i  933 
934 
!  935 
93t 
93' 
93t 

760384 

762129 

763876 

765625 

767376 

769129 

776884 

772641 

774400 

776161 

777924 

779689 

781456 

783225 

784996 

786769 

788544 

790321 

792100 

793881 

795664 

797449 

799236 

801025 

802816 

804609 

806404 

808201 

810000 

811801 

813604 

815409 

817216 

819025 

820836 

822649 

824464 

826281 

828100 

829921 

831744 

833569 

835396 

837225 

839056 

840889 

842724 

844561 

846401 

848241 

850081 

851929 

853776 

855625 

857476 

859329 

861184 

863041 

86490C 

86676] 

868624 

870489 

872356 

874225 

876096 

877969 

87984' 

663054848 

665338617 

667627624 

669921875 

1-72221376 

674526133 

676836152 

679151439 

681472000 

683797841 

686128968 

688465387 

690807104 

693154125 

695506456 

697864103 

700227072 

702595369 

704969000 

707347971 

709732288 

712121957 

714516984 

716917375 

719323136 

721734273 

724150792 

726572699 

729000000 

731432701 

733870808 

736314327 

738763264 

741217625 

743677416 

746142643 

748613312 

751089429 

753571000 

756058031 

758550528 

761048497 

763551944 

766060875 

768575296 

771095213 

773620632 

776151559 

77868800C 

781229961 

783777448 

786330467 

788889024 

791453125 

794022776 

796597983 

79917875* 

801765089 

804357o(X 

806954491 

809557568 

812166237 

814780504 

817400375 

820025856 

822656955 

825293672 

29-5296461 

29-5465734 

29-5634910 

29-5803989 

29-5972972 

29-6141858 

29-6310648 

29-6479342 

29-6647939 

29-6816442 

29.6984843 

29-7153159 

29-7321375 

29-7489496 

29-7657521 

29-7825452 

29-7993289 

29-8161030 

29-8328678 

29-8496231 

29-8663690 

29-8831056 

29-8998328 

29-9165506 

29-9332591 

29-9499583 

29-9666481 

29- 9833287 
30  0000000 

30- 0166620 
30-0333148 
30-0499534 
30-0665928 
30-0832179 
30-0998339 
30-1164407 
30-1330383 
30-1496269 
30-1662063 
30-1827765 
30-1993377 
30-2158399 
30-2324323 
30-2489663 
30-2654919 
30-2820079 
30-2985148 
30-3150128 
30-3315018 
30-3479816 
30-3644529 
30  3809151 
30-397363: 
30-413812' 
30-430243] 
30-446674 ’ 
30-4630924 
30-4795011 
30-4959014 
30-5122921 
30-5286751 
30-545048' 
30-5614131 
30-577769’ 
30-594117] 
30-610455' 
30-626785' 

9553712 

9-557363 

9-561011 

9-564656 

9-568298 

9-571938 

9-575574 

9-579208 

9-582840 

9-586468 

9-590094 

9-593717 

9-597337 

9-600955 

9-604570 

9-608182 

9-611791 

9-615398 

9-619002 

9-622603 

9-626202 

9-629797 

9-633591 

9-636931 

9-640569 

9-644151 

9-647737 

9-651317 

9-654394 

9-658463 

9-662040 

9-665610 

9-669176 

9-672740 

9-676302 

9-679860 

9-683417 

9636970 

9-690521 

9-694069 

9-697615 

9-701158 

9-701699 

9-708237 

9-711772 

9-715305 

9-718835 

9-722363 

9-725838 

9-729411 

9  732931 
9-736448 
9-739963 
9-743176 
9-746986 
9-750493 
9-753998 
9-757500 
9-761000 
9-764497 
9-767992 
9-771484 
9-774974 
9-778462 
9  781947 
9-785429 
9-788909 

APPENDIX. 


21 


No. 

Square. 

Cube. 

Sq.  Root. 

CubeRoot. 

No. 

Square. 

Cube. 

Sq.  Root. 

CubeRoot. 

939 

881721 

827936019 

30-6431069 

9  792386 

970 

940900 

912673000 

31- 1448230- 

9-898983 

910 

883600 

830584000 

30  6594194 

9  795861 

971 

912841 

915498611 

31 -160872s 

9 -9023  83 

941 

885481 

833237621 

30-6757233 

9-799334 

972 

944784 

918330048 

31-1769145 

9-905782 

942 

8S7364 

835896888 

30-6920185 

9-802804 

973 

946729 

921167317 

31-1929479 

9-909178 

943 

889249 

838561807 

30-7083051 

9-806271 

974 

948676 

924010424 

31-2089731 

9-912571 

944 

891136 

841232334 

30-7245830 

9-809736 

975 

950625 

926859375 

31-2249900 

9-915962 

945 

893025 

843908625 

30-7408523 

9-813199 

976 

952576 

929714176 

31-2409987 

9-919351 

946 

894916 

846590536 

30-7571130 

9-816659 

977 

954529 

932574833 

31-2569992 

9-922738 

947 

896809 

849278123 

30-7733651 

9-8201 17 

978 

956484 

935441352 

31-2729915 

9-926122 

948 

898704 

851971392 

30-7896086 

9-823572 

979 

958441 

938313739 

31-288J757 

9-929504 

949 

900601 

854070349 

30-8058436 

9-827025 

980 

960400 

941192000 

31-3049517 

9-932834 

950 

902500 

857375000 

30-8220700 

9-830476 

981 

962361 

944076141 

31-3209195 

9-936261 

951 

904101 

860085351 

30-8382879 

9-833924 

982 

964324 

946966168 

31-3368792 

9-939636 

952 

906304 

862801408 

30-8544972 

9-837369 

983 

966289 

949862087 

31-3528308 

9-943009 

953 

908209 

865523177 

30-8706981 

9-840813 

984 

968256 

952763904 

31-3687743 

9-916380 

954 

910116 

868250664 

30-8868904 

9-844254 

985 

970225 

955671625 

31-3847097 

9-949748 

955 

912025 

870983875 

30-9030743 

9-847692 

986 

972196 

958535256 

31-4006369 

9-953114 

956 

913936 

873722816 

30-9192497 

9-851128 

987 

974169 

961504803 

31-4165561 

9956477 

957 

915849 

876467493 

30-9354166 

9-854562 

288 

976144 

964430272 

31-4324673 

9-959839 

958 

917764 

879217912 

30-9515751 

9-857993 

989 

978121 

967361669 

31-4483704 

9-963198 

959 

919681 

881974079 

30-9677251 

9-861422 

990 

980100 

970299000 

31-4642654 

9-966555 

960 

921600 

884736000 

30-9838668 

9-864848 

991 

982081 

973242271 

31-4801525 

9-969909 

961 

923521 

887503681 

31-0000000 

9-868272 

992 

984064 

976191488 

31-4960315 

9-973262 

962 

925444 

890277128 

31-0161248 

9-871694 

993 

986049 

979146657 

31-5119025 

9-976612 

963 

927369 

893056347 

31-0322413 

9-875113 

994 

988036 

982107784 

31-5277655 

9-979960 

964 

929296 

895841314 

31-0483494 

9-878530 

995 

990025 

985071875 

31-5436206 

9-983305 

965 

931225 

898632125 

31  0644491 

9-881945 

996 

992016 

988047936 

31-5594677 

9-986649 

966 

933156 

901428696 

31-0805405 

9-885357 

997 

994009 

991026973 

31-5753068 

9-989990 

967 

935089 

904231063 

310966236 

9-888767 

998 

996004 

991011992 

31-5911380 

9-993329 

968 

937024 

907039232 

31  1126984 

9-892175 

999 

998001 

997002999 

31-6069613 

9-996666 

969 

938961 

909853209 

31-1287648 

9-895580 

1000 

1000000  1000000000 

31-6227766 

10000000 

The  following  rules  are  for  finding  the  squares,  cubes  and  roots,  of 
numbers  exceeding  1,000. 

To  find  the  square  of  any  number  divisible  without  a  remainder. 
Rule. — Divide  the  given  number  by  such  a  number,  from  the  forego¬ 
ing  table,  as  will  divide  it  without  a  remainder  ;  then  the  square  of  the 
quotient,  multiplied  by  the  square  of  the  number  found  in  the  table, 
will  give  the  answer. 

Example. — What  is  the  square  of  2,000  ?  2,000,  divided  by  1,000, 

a  number  found  in  the  table,  gives  a  quotient  of  2,  the  square  of  which 
is  4,  and  the  square  of  1,000  is  1,000,000,  therefore  : 

4  X  1,000,000  =  4,000,000  :  the  Ans. 

Another  example. — What  is  the  square  of  1,230  ?  1,230,  being  di¬ 

vided  by  123,  the  quotient  will  be  10,  the  square  of  which  is  100,  and 
the  square  of  123  is  15,129,  therefore  : 

100  X  15,129  -=  1,512,900:  the  Ans. 

To  find  the  square  of  any  number  not  divisible  without  a  remainder. 
Rule. — Add  together  the  squares  of  such  two  adjoining  numbers,  from 
the  table,  as  shall  togeiher  equal  the  given  number,  and  multiply  the 
sum  by  2 ;  then  this  product,  less  1,  will  be  the  answer. 

Example. — What  is  the  square  of  1,487  ?  The  adjoining  numbers 
743  and  744,  added  together,  equal  the  given  number,  1,487,  and  the 
square  of  743  =  552,049,  the  square  of  744  =  553,536,  and  these 
added,  =  1,105,585,  therefore  : 

1,105,585  x  2  —  2,211,170  —  1  —  2,211,169 :  the  Ans. 

To  find  the  cube  of  any  number  divisible  without  a  remainder. 
Rule. — Divide  the  given  number  by  such  a  number,  from  the  forego- 


22 


APPENDIX. 


ing  table,  as  will  divide  it  without  a  remainder  ;  then,  the  cube  of  the 
quotient,  multiplied  by  the  cube  of  the  number  found  in  the  table,  will 
give  the  answer. 

Example. — What  is  the  cube  of  2,700  ?  2,700,  being  divided  by  900, 
the  quotient  is  3,  the  cube  of  which  is  27,  and  the  cube  of  900  is 
729,000,000,  therefore  : 

27  X  729,000,000  —  19,683,000,000:  the  Ans. 

To  find  the  square  or  cube  root  of  numbers  higher  than  is  found  in  the 
table.  Rule. — Select,  in  the  column  of  squares  or  cubes,  as  the  case 
may  require,  that  number  which  is  nearest  the  given  number  ;  then 
the  answer,  when  decimals  are  not  of  importance,  will  be  found  di¬ 
rectly  opposite  in  the  column  of  numbers. 

Example. — What  is  the  square-root  of  87,620  ?  In  the  column  of 
squares,  87,616  is  nearest  to  the  given  number  ;  therefore,  296,  im¬ 
mediately  opposite  in  the  column  of  numbers,  is  the  answer,  nearly. 

Another  example. — What  is  the  cube-root  of  110,591  ?  In  the  co¬ 
lumn  of  cubes,  110,592  is  found  to  be  nearest  to  the  given  number; 
therefore,  48,  the  number  opposite,  is  the  answer,  nearly. 

To  find  the  cube-root  more  accurately.  Rule. — Select,  from  the  co¬ 
lumn  of  cubes,  that  number  which  is  nearest  the  given  number,  and 
add  twice  the  number  so  selected  to  the  given  number  ;  also,  add  twice 
the  given  number  to  the  number  selected  from  the  table.  Then,  as 
the  former  product  is  to  the  latter,  so  is  the  root  of  the  number  selected 
to  the  root  of  the  number  given. 

Example. — What  is  the  cube-root  of  9,200  ?  The  nearest  number 
in  the  column  of  cubes  is  9,261,  the  root  of  which  is  21,  therefore  : 

9261  9200 

2  2 


18522  18400 

9200  9261 


As  27,722  is  to  27,661,  so  is  21  to  20-953  -f-  the  Ans. 

21 


27661 

55322 


27722)580881(20-953  + 
55444 


264410 

249498 


149120 

138610 


105100 

83166 


21934 


APPENDIX. 


23 


To  find,  the  square  or  cube  root  of  a  whole  number  with  decimals. 
Rule. — Subtract  the  root  of  the  whole  number  from  the  root  of  the  next 
higher  number,  and  multiply  the  remainder  by  the  given  decimal ; 
then  the  product,  added  to  the  root  of  the  given  whole  number,  will 
give  the  answer  correctly  to  three  places  of  decimals  in  the  square- 
root,  and  to  seven  in  the  cube-root. 

Example. — What  is  the  square-root  of  11*14  ?  The  square-root  of 
11  is  3*3166,  and  the  square-root  of  the  next  higher  number,  12,  is 
3*4641,  therefore  : 

3*4641 

3-3166 


•1475 

•14 


5900 

1475 


•020650 

3-3166 


3-33725 :  the  Ans. 


RULES  FOR  THE  REDUCTION  OF  DECIMALS. 

To  reduce  a  fraction  to  its  equivalent  decimal.  Rule. — Divide  the 
numerator  by  the  denominator,  annexing  cyphers  as  required. 
Example. — What  is  the  decimal  of  a  foot  equivalent  to  3  inches  ? 

3  inches  is  T3?  of  a  foot,  therefore  : 

T®5  ...  12)  3  00 

•25  Ans. 

Another  example. — What  is  the  equivalent  decimal  of  f  of  an  inch  1 
*  ....  8)  7-000 


•875  Ans. 

To  reduce  a  compound  fraction  to  its  equivalent  decimal.  Rule. — In 
accordance  with  the  preceding  rule,  reduce  each  fraction,  commen¬ 
cing  at  the  lowest,  to  the  decimal  of  the  next  higher  denomination,  to 
which  add  the  numerator  of  the  next  higher  fraction,  and  reduce  the 
sum  to  the  decimal  of  the  next  higher  denomination,  and  so  proceed  to 
the  last ;  and  the  final  product  will  be  the  answer. 

Example. — What  is  the  decimal  of  a  foot  equivalent  to  5  inches,  | 
and  TV  of  an  inch  ? 

The  fractions*  in  this  case  are,  £  of  an  eighth,  f  of  an  inch,  and 
of  a  foot,  therefore : 


24 


APPENDIX. 


* . 2)  1*0 

•5 

3*  eighths. 

a. .  8)  3-5000 

•4375 

5-  inches. 


i 


JL. 

1  2 


12)  5-437500 


•453125  Ans. 

The  process  may  be  condensed,  thus  ;  write  the  numerators  of  the 
given  fractions,  from  the  least  to  the  greatest,  under  each  other,  and 
place  each  denominator  to  the  left  of  its  numerator,  thus  : 

*  ....  2  1-0 


JL 

2 


a. 

s 


8 


3-5000 


* 


12 


5-437500 


•453125  Ans. 

To  reduce  a  decimal  to  its  equivalent  in  terms  of  lower  denominations 4 
Rule. — Multiply  the  given  decimal  by  the  number  of  parts  in  the  next 
less  denomination,  and  point  off  from  the  product  as  many  figures  at 
the  right  hand,  as  there  are  in  the  given  decimal ;  then  multiply  the 
figures  pointed  off,  by  the  number  of  parts  in  the  next  lower  denomina¬ 
tion,  and  point  off  as  before,  and  so  proceed  to  the  end ;  then  the  seve¬ 
ral  figures  pointed  off  at  the  left  will  be  the  answer. 

Example. — What  is  the  expression  in  inches  of  0-390625  feet  ? 

Feet  0-390625 

12  inches  in  a  foot. 


Inches  4-687500 

8  eighths  in  an  inch. 


Eighths  5-5000 

2  sixteenths  in  an  eighth 


Sixteenth  1-0 


Ans.,  4  inches  f  and  T’T. 

Another  example. — What  is  the  expression,  in  fractions  of  an  inch, 
of  0-6875  inches  ? 

Inches  0-6875 

8  eighths  in  an  inch. 


Eighths  5-5000 

2  sixteenths  in  an  eighth. 

Sixteenth  1-0 

Ans.,  |  and  tV- 


TABLE  OF  CIRCLES 


(From  Gregory’s  Mathematics.) 

From  this  table  may  be  found  by  inspection  the  area  or  circumfe¬ 
rence  of  a  circle  of  any  diameter,  and  the  side  of  a  square  equal  to  the 
area  of  any  given  circle  from  1  to  100  inches,  feet,  yards,  miles,  &c. 
If  the  given  diameter  is  in  inches,  the  area,  circumference,  &c.,  set 
opposite,  will  be  inches  ;  if  in  feet,  then  feet,  &c. 


Diam. 

Area. 

Circum. 

Side  of 
equal  sq. 

Diam. 

Area. 

Circum. 

Side  of 
equal  sq. 

■25 

•04908 

•78539 

•22155 

•75 

90-76257 

33-77212 

9-52693 

•5 

•19635 

1-57079 

•44311 

11- 

9503317 

34-55751 

9*74849 

•75 

■44178 

2-35619 

•66467 

•25 

99-40195 

35-34291 

9-97005 

1- 

•78539 

314159 

•88622 

•5 

103-86890 

36  12831 

10-19160 

•25 

1-22718 

3-92699 

M0778 

•75 

108-43403 

36-91371 

10-41316 

•5 

1-76714 

4-71238 

1-32934 

12- 

113-09733 

37-69911 

10-63472 

•75 

2-40528 

5-49778 

1-55089 

•25 

117-85881 

38-48451 

10-85627 

2- 

3-14159 

6-28318 

1-77245 

•5 

122-71846 

39-26990 

11-07783 

•25 

3-97607 

7-06858 

1-99401 

•75 

127-67628 

40-05530 

11-29939 

•5 

4-90873 

7-85393 

2-21556 

13- 

132-73228 

40-84070 

11-52095 

•75 

5-93957 

8-63937 

2-43712 

•25 

137-88646 

41-62610 

11-74250 

3- 

7-06858 

9-42477 

2-65868 

•5 

143-13881 

42-41150 

11-96406 

•25 

8-29576 

10-21017 

2-88023 

•75 

148-48934 

43-19689 

12-18562 

•5 

9-62112 

10-99557 

3-10179 

14- 

153-93804 

43-98229 

12-40717 

•75 

11-04466 

11-78097 

3-32335 

•25 

159-48491 

44-76769 

12-62873 

4- 

12-56637 

12-56637 

3-54490 

•5 

165  12996 

45-55309 

12-85029 

•25 

14-18625 

13-35176 

3-76646 

•75 

170-87318 

46-33349 

13-07184 

•5 

15-90431 

1413716 

3-98802 

15- 

176-71458 

47-12388 

13-29340 

•75 

17-72054 

14-92256 

4-20957 

•25 

182-65418 

47-90928 

13-51496 

5- 

19-63495 

15-70796 

4-43113 

•5 

188-69190 

48-69468 

13-73651 

•25 

21-64753 

16-49336 

4-65269 

•75 

194-82783 

49-48003 

13-95807 

•5 

23-75829 

17-27875 

4-87424 

16- 

201-06192 

50-26548 

14-17963 

*75 

25-96722 

18-06415 

5-09580 

•25 

207-39420 

51-05088 

14-40118 

6- 

28-27433 

18-84955 

5-31736 

•5 

213-82464 

51-83627 

14-62274 

•25 

30-67961 

19-63495 

5-53891 

•75 

220-35327 

52-62167 

14-84430 

•5 

33-18307 

20-42035 

5-76047 

17- 

226-98006 

53-40707 

15-06535 

•75 

35-78470 

21-20575 

5-98203 

•25 

233-70504 

54-19247 

15-28741 

7- 

33-48456 

21-99114 

6-20358 

•5 

240-52818 

54-97737 

15-50897 

•25 

41-28249 

22-77654 

6-42514 

•75 

247-44950 

55-76326 

15-73052 

•5 

44-17864 

23-55194 

6-64670 

18- 

264-46900 

56-54866 

15-95208 

•75 

47-17297 

24-34734 

6-86825 

•25 

266-58667 

57-33406 

16-17364 

8- 

50-26548 

25-13274 

7-08981 

•5 

288-80252 

58-11946 

16-39519 

•25 

53-45616 

25-91813 

7-31137 

•75 

276-11654 

58-90486 

16-61675 

•5 

56-74501 

26-70353 

7-53292 

19- 

283-52873 

59-69026 

16-83831 

•75 

60  13204 

27-48893 

7-75448 

•25 

291-03910 

60-47565 

17-05986 

9- 

63-61725 

28-27433 

7-97604 

•5 

298-64765 

61-26105 

17-28142 

•25 

67-20063 

29-05973 

8-19759 

•75 

306-35437 

62-04645 

17-50298 

•5 

70-88218 

29-84513 

8-41915 

20- 

314-15926 

62-83185 

17-72453 

•75 

74-66191 

30-63052 

8-64071 

•25 

322-06233 

63-61725 

17-94609 

10- 

78-53981 

31-41592 

8-86226 

•5 

330-06357 

64-40264 

18-16765 

•25 

82-51589 

32-20132 

9-08382 

•75 

338-16299 

65-18804 

18-38920 

•5 

86 -59014  j 

32-98672 

9-30538 

21- 

346-36059 

65-97344 

18-61076 

4* 


26 


APPENDIX 


Diam. 

Area. 

Circum. 

Side  of 
equal  sq. 

Diam. 

Area. 

Circum. 

Side  of 
equal  sq. 

21-25 

354-65635 

66-75884 

18-83232! 

38- 

1134-11494, 

119-38052 

33-67662 

•5 

363-05030 

67-54424 

19-05387 

•25 

1149-08660 

120-16591 

33-89817 

•75 

371-54241 

68-32964 

19-27543 

•5 

1164-15642 

120-95131 

34  11973 

22- 

380' 13271 

69-115031 

19-49699 

•75 

1179-32442 

121-73671 

34-34129 

•25 

388-82117 

69-90043 

19-71854 

39- 

1194-59060 

122-5221 1 

34-56285 

•5 

397-60782 

70-68583 

19-94010 

•25 

1209-95495 

123-30751 

34-78440 

•75 

406-49263 

71-47123 

20-16166 

•5 

1225-41748 

124-09290 

3500596 

23- 

415-47562 

72-25663 

20-38321 

•75 

1240-97818 

124-87830 

35-22752 

•25 

424-55679 

73-0420& 

20-60477 

40- 

1256-63704 

125-66370 

35-44907 

•5 

43373613 

73-82742 

20-82633; 

■25 

1272-39411 

126-44910 

35-67063 

•75 

443-01365 

74-61282 

21  04788 

■5 

1288-24933 

12723450 

35-89219 

24- 

452-38934 

75-39822 

21-26944 

■75 

1304-20273 

128-01990 

36  11374 

•25 

461-86320 

76-18362 

21-49100 

41- 

1320-25431 

128-80529 

36-33530 

•5 

471-43524 

76-96902 

21-71255 

•25 

1336-40406 

129-59069 

3655686 

•75 

48M0546 

77-75441 

21-93411 

•5 

1352-65198 

130-37609 

36-77841 

25- 

490-87385 

78-53981 

22-15567 

•75 

1368-99808 

13116149 

36-99997 

•25 

500-74041 

79-32521 

22-37722 

42- 

1385-44236 

131-94689 

37-22153 

•5 

510-70515 

80-11061 

22-59878 

•25 

1401  98480 

132-73228 

37-44308 

•75 

520-76806 

80-89601 

22-82034 

•5 

1418-62543 

133-51768 

37  66464 

26- 

530-92915 

81-68140 

23-04190 

•75 

1435-36423 

134-30308 

37-83620 

■25 

541-18842 

82-46680 

23-26345 

43- 

1452-20120 

13508348 

38  10775 

•5 

551-54586 

83-25220 

23-48501 

•25 

1469  13635 

135-87383 

38-32931 

•75 

562-00147 

84-03760 

23-70657 

•5 

1486-16967 

136-65928 

38  55087 

27- 

572-55526 

84-82300 

23-92812 

•75 

1503-30117 

137-44467 

38-77242 

•25 

583-20722 

85-60839 

24  14968 

44- 

1520-53084 

138-23007 

38-99398 

•5 

593-95736 

86-39379 

24-37124 

•25 

1537-85869 

13901547 

39-21554 

•75 

604-80567 

87-17919 

24-59279 

•5 

1556-23471 

139-80087 

39-43709 

28- 

615-75216 

87-96459 

24-81435 

•75 

1572-80890 

140-58627 

39  65365 

•25 

626-79682 

88-74999 

25-03591 

45- 

1590-43128 

14P37166 

39-88021 

•5 

637-93965 

89-53539 

25-25746 

•25 

1608  15182 

142  15706 

4010176 

•75 

649-18066 

90-32078 

25-47902 

•5 

1625-97054 

142-94246 

40-32332 

29- 

660-51985 

91-10618 

25-70058 

•75 

1613  88744 

143-72786 

40  54488 

•25 

671-95721 

91-89158 

25-92213 

46- 

1661-90251 

144  51326 

40-76643 

•5 

683-49275 

92-67698 

26-14369 

•25 

16S001575 

145-29866 

40-98799 

•75 

695-12646 

93-46238 

26-36525 

•5 

1698-22717 

146-08405 

41-20955 

30- 

706-85834 

94-24777 

26-58680 

•75 

1716-53677 

146-86945 

41-43110 

•25 

718  68840 

9503317 

26-80836 

47- 

1734-94454 

14765485 

41-65266 

•5 

730-61664 

95-81857 

27-02992 

•25 

1753-45048 

148-44025 

41  87422 

•75 

742  64305 

96-60397 

27-25147 

•5 

1772-05460 

149-22565 

42-09577 

31- 

751-76763 

97-38937 

27-47303 

•75 

1790-75689 

150-01104 

4231733 

•25 

766-99039 

98-17477 

27-69459 

48- 

1809-55736 

150-79644 

42-53889 

•5 

779-31132 

98-96016 

27-91614 

•25 

1828-45601 

151-58184 

42-76044 

•75 

791-73043 

99-74556 

28-13770 

•5 

1847-45282 

152-36724 

42-98200 

32- 

804-24771 

100-53096 

28-35926 

•75 

1866-54782 

153  15264 

43-20356 

•25 

816-86317 

101-31636 

28-58081 

49- 

1885-74099 

15393804 

43  4251 1 

•5 

829-57681 

102-10176 

28-80237 

•25 

1905-83233 

154-72343 

43-64667 

•75 

842-38861 

102-88715 

29  02393 

•5 

1924-42184 

155-50883 

43-86823 

33- 

855-29859 

103-67255 

29-24548 

•75 

1943-90954 

156-29423 

44-08978 

•25 

868-30675 

104-45795 

29-46704 

50- 

1963-49540 

157-07963 

44-31134 

•5 

881-41308 

105-24335 

29-68860 

•25 

1983-17944 

157-96503 

44-53290 

•75 

894-61759 

106-02875 

29-91015 

•5 

2002-96166 

158-65042 

44-75445 

34- 

907-92027 

106-81415 

30-13171 

•75 

2022-84205 

159-43582 

44-97601 

•25 

921-32113 

107-59954 

30-35327 

51- 

2042-82062 

160-22122 

45  19757 

•5 

934-82016 

108-38494 

30-57482 

•25 

2062-89736 

161-00662 

45-41912 

•75 

94841736 

109-17034 

30-79638 

•5 

2083-07227 

161-79202 

45-64068 

35* 

962-11275 

109-95574 

31-01794 

•75 

2103-34536 

162-57741 

45-86224 

•25 

975-90630 

110-74114 

31-23949 

52- 

2123-71663 

163-36281 

46  08380 

•5 

989-79803 

111-52653 

31-46105 

•25 

2144-18607 

164-14821 

46-30535 

■75 

1003-78794 

112-31193 

31-68261 

■5 

2164-75368 

164-93361 

46  52691 

36- 

1017-87601 

11309733 

31-90416 

•75 

2185-41947 

165-71901 

46-74847 

•25 

1032-06227 

113-88273 

32-12572 

53- 

2206-18344 

166-50441 

46-97002 

•5 

1046-34670 

114-66313 

32-34728 

■25 

2227-04557 

167-28980 

47-19158 

•75 

1060-72930 

115-45353 

32-56883 

•5 

2248-00589 

16807520 

47-41314 

37- 

1075-21008 

116-23892 

32-79039 

•75 

2269-06438 

168-86060 

47-63469 

•25 

1089-78903 

117-02432 

3301195 

54- 

2290-22104 

169-64600 

47-85625 

•5 

1104-46616 

117-80972 

33-23350 

•25 

2311-47588 

170-43140 

4S  07781 

•75 

1119-24147 

118-59572 

33-45506 

•5 

2332-82889 

171-21679 

48-29936 

APPENDIX 


27 


Diam. 

Area. 

Circurn. 

Side  of 
equal  sq. 

Diam. 

Area. 

Circum. 

Side  of 
equal  sq. 

54-75 

2354-28008 

172-00219 

48-52092 

71-5 

4015-15176 

224-62337 

63-36522 

55- 

2375-82344 

172-78759 

48-74248 

•75 

4043-27883 

225-40927 

63-58678 

•25 

2397-47698 

173-57299 

48-96403 

72- 

4071-50407 

226-19467 

63-80333 

•5 

2419-22269 

174-35839 

49-18559 

•25 

4099-82750 

226-98006 

64-02989 

•75 

2441-06657 

175-14379 

49-40715 

•5 

4128-24909 

227-76546 

64-25145 

56- 

246300864 

175-92918 

49-62870 

•75 

4156-76886 

22855686 

64-47300 

•25 

2185-04887 

176-71458 

49-85026 

73- 

4185-33681 

229-33626 

64-69456 

•5 

2507-18728 

177-49998 

50-07182 

•25 

4214  10293 

230-12166 

64-91612 

•75 

2520-42337 

178-28538 

50-29337 

•5 

4242-91722 

230-90706 

65  13767 

57- 

2551-75363 

179-07078 

50-51493 

•75 

4271-82969 

231-69245 

65-35923 

•25 

2574-19156 

179-85617 

50-73649 

74- 

4300-84034 

232-47785 

65-58079 

•5 

2596-72267 

180-64157 

50-95804 

•25 

4329-94916 

233-26325 

65-80234 

•75 

2619-35196 

181-42697 

51-17960 

•5 

4359-15615 

231-04865 

66-02390 

58- 

2642-07942 

182  21237 

51-40116 

•75 

4388-46132 

231-83405 

66-24546 

•25 

2664  90505 

182-99777 

51-62271 

75- 

4417-86466 

235-61944 

66-46701 

•5 

2687-82886 

183-78317 

51  84427 

■25 

4447-36618 

236-40484 

66-68857 

•75 

2710-85084 

184-56856 

52-06583 

•5 

4476-96538 

237-19024 

66-91043 

59- 

2733-97100 

185-35336 

52-28738 

•75 

4506-66374 

237-97564 

67-13168 

•25 

2757-18933 

186  13936 

52-50894 

76- 

4536-45979 

238-76104 

67-35324 

•5 

2780-50584 

186-92476 

52-73050 

•25 

4566  35400 

239-54613 

67-57480 

•75 

2803  92053 

187-71016 

52-95205 

•5 

4596  34640 

240-33183 

67-79635 

60- 

2327-43338 

188-49555 

53-17364 

•75 

4626  43696  24M1723 

63-01791 

•25 

2851-04442 

189-28095 

53  39517 

77- 

4656-62571)  241-90263 

68-23947 

•5 

2874-75362 

190-06635 

53-61672 

•25 

4686-91262  242-68803 

68-46102 

•75 

2898-56100 

190-85175 

53-83828 

■5 

4717-29771 

243-47343 

68  68258 

61- 

2922-46656 

191-63715 

54-05984! 

•75 

4747-78098 

244-25382 

68-90414 

•25 

2946-47029 

192-42255 

54-28139 

78- 

4778-36242 

245  04422 

69-12570 

•5 

2970-57220 

193-20794 

54-502951 

•25 

4809-04204 

245-82962 

69  31725 

*75 

2994-77228 

193-99331 

54-7245 1| 

•5 

4839-81983 

246-61502 

69-56881 

G2- 

3019-07054 

194-77874 

54-94606! 

•75 

4870-79579 

247-40042 

69-79037 

•25 

304  3  46697 

195-56414 

55-16762 

79- 

4901-66993 

248-18581 

70-01192 

•5 

3067-96157 

196-34954 

55-38918 

•25 

4932-74225 

248-97121 

70-23318 

■75 

3092-55435 

197-13493 

55-61073 

•5 

4963-91274 

249-75661 

70-45504 

63- 

3117-24531 

197-92033 

55-83229 

•75 

4995-18140 

250-34201 

70-67659 

•25 

314203444 

198-70573 

56-05385 

80- 

5026-54824 

251-32741 

70-89815 

•5 

3166-92174 

199-49113 

56-27540) 

•25 

5058-01325 

252-11281 

71-11971 

•75 

3191-90722 

200-27653 

56-49696 

•5 

5089-57644 

252-89820 

71-34126 

64- 

3216-99087 

201-06  W2 

56-71852 

•75 

5121-23781 

253-63360 

71-56282 

•25 

3242-17270 

201-84732 

56-94007 

81- 

5152-99735 

254-46900 

71-78438 

•5 

326745270 

20263272 

57-16163 

•25 

5184-85506 

255-25440 

72-00593 

•75 

3292  83088 

203-41812 

57-38319 

•5 

5216-81095 

256-03J80 

72-22749 

65- 

3318-30724 

204-20352 

57-60475 

•75 

5248-86501 

256-82579 

72-44905 

•25 

3313-88176 

294-98892 

57-826301 

82- 

5281-01725 

257-61059 

'2-67060 

•5 

3369-55447 

205-77431 

5804786 

•25 

5313  26766 

258-39599 

72-89216 

•  -75 

3395  32534 

206-55971 

58-26942 

•5 

5345-61624 

259-18139 

73-11372 

66- 

3421-19439 

207-34511 

58-49097 

•75 

5378-06301 

259-96679 

73-33527 

•25 

3447-16162 

208-13051 

58-71253 

83- 

5410-60794 

260-75219 

73-55683 

•5 

347322702 

208-91591 

58-93409 

•25 

5443-25105 

261-53758 

73-77839 

•75 

3499-39060 

209-70130 

59-15564 

•5 

5475-99234 

262-32298 

73-99994 

07- 

3525-65235 

21048670 

59-37720 

■75 

5508-83180 

263-10838 

74-22150 

25 

3552-01228 

211-27210 

59-59876 

84- 

5541-76944 

263-89378 

74-44306 

•5 

3578-47033 

212-05750 

59-82031 

•25 

5574-80525 

264-67918 

74-66461 

•75 

360502665 

212-84290 

60-04187 

•5 

5607-93923 

265-46457 

74-S8617 

68- 

3631-68110 

213-62^30 

60-26343 

•75 

5641  •  17 1 39) 

266-24997 

75-10773 

•25 

3658-43373 

214-41369 

60-48498 

85- 

5674-50173 

267-03537 

75-32923 

•5 

3685-28453 

215-19909 

60-70654 

•25 

5707-93023! 

267-82077 

75-55084 

•75 

3712-23350 

215-98149 

60-92810 

•5 

5741-45632 

268-60617 

75  77240 

69' 

3739-28065 

216  76989 

6114965 

•75 

5775-08178 

269-39157 

75-99395 

•25 

3766-42597 

217-55529 

61-37121 

86- 

5808-804811 

270-17696 

7621551 

•5 

379366947 

218  34068 

61-59277 

•25 

5842-62602 

270-96236 

7643707 

•75 

3321  01115 

219-12608 

61-81432 

•5 

5876-54540 

271-74776 

76-65362 

70- 

3848-45100 

219-91143 

62-03588 

•75 

5910-56296 

272-53316 

76-88018 

•25 

3875-98902 

220-69683 

62-25744 

87- 

5944  67869 

273-31856 

7710174 

■5 

3303-62522 

221-48228 

62-47899 

•25 

5978-89260 

274-10395 

77-32329 

•75 

3931-35959 

222-26768 

62-70055 

•5 

6013-20468 

274-88935 

77-54485 

71- 

3959-19214 

223-05307 

62-92211 

•75 

6047-61494 

275-67475 

77-766-41 

•25 

3987-12286 

223-83847 

63-14366 

88- 

6082-12337 

276-46015 

77-98796 

28 


APPENDIX. 


Diam. 

Area. 

Circum. 

Side  of 
equal  sq. 

Diam. 

Area. 

Circum. 

Side  of 
equal  sq. 

88-25 

6116  72998 

277-24555 

78-20952 

94-25 

6976-74097 

2.  6-01510 

83-52688 

•5 

6151-43476 

27803094 

78-43103 

•5 

7013-80194 

296-88050 

83-74344 

•75 

6186-23772 

278-81634 

78-65263 

•75 

7050-96109 

297  66590 

83-97000 

89- 

622 1-13885 

279-60174 

78-87419 

95- 

7033-21342 

298-45130 

84-19155 

•25 

6256-13315 

230-33714 

79-09575 

•25 

7125-57992 

299-23670 

84-41311 

•5 

6231-23563 

23M7254 

79-3173-i 

•5 

7163-02759 

300  02209 

84-63467 

•75 

6326-43129 

281-95794 

79-53836 

•75 

7200-57944 

300-80749 

84-85622 

90- 

6361-72512 

282-74333 

79-76042 

96- 

7233-22947 

30159239 

85-07778 

•25 

6397-11712 

283-52873 

79-98193 

•25 

7275-97767 

302-37829 

85-29934 

•5 

6432-60730 

284-31413 

80-20353 

•5 

7313-82404 

303-16369 

85-52089 

•75 

6463- 19566 

285-09953 

80-42509 

•75 

7351-76859 

303-94908 

85-74245 

91- 

6503-88219 

285-83493 

80-64669 

97- 

7389-81131 

304-73448 

85-96401 

.25 

6533-66689 

286-67032 

80-86820 

•25 

7427-95221 

30551983 

86-18556 

•5 

6575-54977 

287-45572 

81-03976 

*5 

7466-19129 

306-30528 

86-40712 

•75 

6611-53082 

288-24112 

81-31132 

-75 

7504-52853 

307-09068 

86-62368 

92- 

6647-61005 

289-02652 

81-53287 

98- 

7542-96396 

307-87603 

86-85023 

•25 

6633-78745 

289-81192 

81-75443 

•25 

7581-49755 

308-66147 

87-07179 

•5 

6720-06303 

290-59732 

81-97599 

■5 

7620-12933 

309-44687 

87-29335 

•75 

6756-43678 

291-33271 

82  19754 

•75 

7653-85927 

310-23227 

87-51490 

93- 

6792-90871 

292-16811 

82-41910 

99- 

7697-68739 

31101767 

87-73646 

•25 

6829-47831 

292-95351 

82-64066 

•25 

7736-61369 

311-80307 

87-95802 

•5 

6866  14709 

293-73391 

82-86221 

•5 

7775-63816 

312-58846 

88-17957 

•75 

6902-91354 

294-52431 

83-03377 

•75 

7814-76031 

313-37336 

88-40113 

94- 

6939-77817 

295-30970 

83-30533 

100- 

7853-98163 

314-15926 

88-62269 

The  following  rules  are  for  extending  the  use  of  the  above  table. 

To  find  the  area,  circumference,  or  side  of  equal  square,  of  a  circle 
having  a  diameter  of  more  than  100  inches,  feet,  fyc.  Rule. — Divide 
the  given  diameter  by  a  number  that  will  give  a  quotient  equal  to  some 
one  of  the  diameters  in  the  table  ;  then  the  circumference  or  side  of 
equal  square,  opposite  that  diameter,  multiplied  by  that  divisor,  or,  the 
area  opposite  that  diameter,  multiplied  by  the  square  of  the  aforesaid 
divisor,  will  give  the  answer. 

Example. — What  is  the  circumference  of  a  circle  whose  diameter  is 
228  feet  ?  228,  divided  by  3,  gives  76,  a  diameter  of  the  table,  the  cir¬ 
cumference  of  which  is  238-761,  therefore  : 

238-761 

3 


716-283  feet.  Ans. 

Another  example. — What  is  the  area  of  a  circle  having  a  diameter 
of  150  inches  ?  150,  divided  by  10,  gives  15,  one  of  the  diameters  in 
the  table,  the  area  of  which  is  176-71458,  therefore  : 

176-71458 

100  =-  10  X  10 


17,671-45800  inches.  Ans. 

To  find  the  area,  circumference,  or  side  of  equal  square,  of  a  circle 
having  an  intermediate  diameter  to  those  in  the  table.  Rule. — Multiply 
the  given  diameter  by  a  number  that  will  give  a  product  equal  to  some 
one  of  the  diameters  in  the  table  ;  then  the  circumference  or  side  of 
equal  square  opposite  that  diameter,  divided  by  that  multiplier,  or,  the 
area  opposite  that  diameter  divided  by  the  square  of  the  aforesaid  mul¬ 
tiplier,  will  give  the  answer. 


APPENDIX. 


29 


Example. — What  is  the  circumference  of  a  circle  whose  diameter  is 
6£,  or  6-125  inches  1  6-125,  multiplied  by  2,  gives  12-25,  one  of  the 

diameters  of  the  table,  whose  circumference  is  38-484,  therefore  : 

2)38-484 


19-242  inches.  Ans. 

Another  example. — What  is  the  area  of  a  circle,  the  diameter  of 
which  is  3-2  feet  ?  3-2,  multiplied  by  5,  gives  16,  and  the  area  of  16 
is  201-0619,  therefore  : 

5  X  5  —  25)201-0619(8-0424  +  feet.  Ans. 

200 


106 

100 


61 

50 

119 

100 


19 

Note. — The  diameter  of  a  circle,  multiplied  by  3-14159,  will  give 
its  circumference  ;  the  square  of  the  diameter,  multiplied  by  -78539, 
will  give  its  area  ;  and  the  diameter,  multiplied  by  -88622,  will  give 
the  side  of  a  square  equal  to  the  area  of  the  circle. 


TABLE  SHOWING  THE  CAPACITY  OF  WELLS,  CISTERNS,  &C. 


The  gallon  of  the  state  of  New- York  is  required  to  contain  8  pounds  of  pure  water;  and 
since  a  cubic  foot  of  pure  water  weighs  62-5  pounds,  the  gallon  contains  221-184  cubic 
inches.  Upon  these  data  the  following  table  is  computed. 


One  foot  in  depth  of  a  cistern  of 


3  feet 

diameter  will  contain 

-  -  55-223  gallons. 

H 

do. 

do. 

75-164 

do. 

4 

do. 

do. 

-  98-174 

do. 

4* 

do. 

do. 

124-252 

do. 

5 

do. 

do. 

-  153-39 

do. 

5* 

do. 

do. 

185-611 

do. 

6 

do. 

do. 

-  220-893 

do. 

do. 

do. 

259-242 

do. 

7 

do. 

do. 

-  300-66 

do. 

8 

do. 

do. 

392-699 

do. 

9 

do. 

do. 

-  497-009 

do. 

10 

do. 

do. 

613-592 

do. 

12 

do. 

do. 

-  883-573 

do. 

Note. — The  area  of  a  circle  in  feet,  divided  by  the  decimal,  -128, 
will  give  the  number  of  gallons  per  foot  in  depth. 


TABLE  OF  POLYGONS. 


(From  Gregory’s  Mathematics.) 


No.  of 
sides. 

Names. 

Multipliers  for 
areas. 

Radius  of  cir- 
cum.  circle. 

Factors  for 
sides. 

3 

Trigon 

0-4330127 

0-5773503 

1-732051 

4 

Tetragon,  or  Square 

1-0000000 

0-7071068 

1-414214 

5 

Pentagon  - 

1-7204774 

0-8506508 

1-175570 

6 

Hexagon 

2-5980762 

1-0000000 

1-000000 

7 

Heptagon  - 

3-6339124 

1-1523824 

0-867767 

8 

Octagon 

4-8284271 

1-3065628 

0-765367 

9 

Nonagon  - 

6-1818242 

1-4619022 

0-684040 

10 

Decagon 

7-6942088 

1-6180340 

0-618034 

11 

Undecagon 

9-3656399 

1-7747324 

0-563465 

12 

Dodecagon  - 

11-1961524 

1-9318517 

0-517638 

To  find  the  area  of  any  regular  polygon ,  whose  sides  do  not  exceed 
twelve.  Rule. — Multiply  the  square  of  a  side  of  the  given  polygon  by 
the  number  in  the  column  termed  Multipliers  for  areas,  standing  op¬ 
posite  the  name  of  the  given  polygon,  and  the  product  will  be  the  an¬ 
swer.  Example. — What  is  the  area  of  a  regular  heptagon,  whose 
sides  measure  each  2  feet  ? 

3-6339124 

4  =  2x2 


14-5356496:  Ans. 

To  find  the  radius  of  a  circle  which  will  circumscribe  any  regular 
polygon  given,  whose  sides  do  not  exceed  twelve.  Rule. — Multiply  a 
side  of  the  given  polygon  by  the  number  in  the  column  termed  Radius 
of  circumscribing  circle,  standing  opposite  the  name  of  the  given  poly¬ 
gon,  and  the  product  will  give  the  answer.  Example. — What  is  the 
radius  of  a  circle  which  will  circumscribe  a  regular  pentagon,  whose 
sides  measure  each  10  feet  ? 

•8506508 

10 


8-5065080  :  Ans. 

To  find  the  side  of  any  regular  polygon  that  may  be  inscribed  within 
a  given  circle.  Rule. — Multiply  the  radius  of  the  given  circle  by  the 
number  in  the  column  termed  Factors  for  sides,  standing  opposite  the 
name  of  the  given  polygon,  and  the  product  will  be  the  answer.  Ex¬ 
ample. — What  is  the  side  of  a  regular  octagon  that  may  be  inscribed 
within  a  circle,  whose  radius  is  5  feet  ? 

•765367 

5 


3-826835:  Ans. 


WEIGHT  OF  MATERIALS 


Woods. 

lUci'  III  l l 

cubic  foot. 

Metals. 

ujs.  m  a 
cubic  foot. 

Apple, 

-  49 

Wire-drawn  brass, 

-  534 

Ash, 

- 

45 

Cast  brass, 

506 

Beach, 

• 

-  40 

Sheet-copper, 

-  549 

Birch, 

• 

45 

Pure  cast  gold,  - 

-  1210 

Box, 

. 

-  60 

Bar-iron, 

475  to  487 

Cedar, 

- 

28 

Cast  iron,  - 

450  to  475 

Virginian  red  cedar, 

-  40 

Milled  lead,  - 

-  713 

Cherry, 

- 

38 

Cast  lead, 

709 

Sweet  chestnut, 

.  36 

Pewter, 

-  453 

Horse-chestnut, 

- 

34 

Pure  platina, 

-  1345 

Cork, 

-  15 

Pure  cast  silver, 

-  654 

Cypress, 

- 

28 

Steel, 

486  to  490 

Ebony,  - 

*  83 

Tin,  - 

-  456 

Elder, 

- 

43 

Zinc, 

439 

Elm, 

-  34 

Stone,  Earths,  Ac. 

Fir,  (white  spruce,) 

29 

Brick,  Phila.  stretchers,  105 

Hickory, 

-  52 

North  river  common  hard 

Lance-wood, 

- 

59 

brick, 

-  107 

Larch, 

-  31 

Do.  salmon  brick,  100 

Larch,  (whitewood,) 

22 

Brickwork,  about 

95 

Lignum-vitse,  - 

-  83 

Cast  Roman  cement, 

-  100 

Logwood, 

- 

57 

Do.  and  sand  in  equal  parts,  113 

St.  Domingo  mahogany,  -  45 

Chalk, 

144  to  166 

Honduras, or  bay 

mahogany,  35 

Clay,  - 

-  119 

Maple, 

- 

47 

Potter’s  clay, 

112  to  130 

White  oak, 

. 

43  to  53 

Common  earth, 

95  to  124 

Canadian  oak, 

_ 

54 

Flint, 

-  163 

Red  oak, 

- 

-  47 

Plate-glass, 

172 

Live  oak, 

. 

76 

Crown-glass,  - 

-  157 

White  pine, 

- 

23  to  30 

Granite, 

158  to  187 

Yellow  pine, 

- 

34  to  44 

Quincy  granite, 

-  166 

Pitch  pine, 

- 

46  to  58 

Gravel, 

109 

Poplar, 

- 

25 

Grindstone, 

-  134 

Sycamore, 

- 

-  36 

Gvpsum,  (Plaster-stone,)  142 

Walnut, 

- 

40 

Unslaked  lime,  - 

52 

32 


APPENDIX. 


Limestone, 

lbs.  in  a 
cubic  foot. 

-  118  to  198 

Marble, 

161  to  177 

New  mortar,  - 

-  107 

Dry  mortar, 

90 

Mortar  with  hair, 

(Plaster- 

ing>)  - 

-  105 

Do.  dry, 

86 

Do.  do.  including  lath 
and  nails,  from  7  to  11 
lbs.  per  superficial  foot. 
Crystallized  quartz,  -  165 

Pure  quartz-sand,  -  171 

Clean  and  coarse  sand,  100 

Welsh  slate,  -  180 

Paving  stone,  -  -  151 

Pumice  stone,  -  -  56 

Nyack  brown  stone,  -  148 

Connecticut  brown  stone,  170 
Nyack  blue  stone,  -  171 


lbs.  in  a 
cubic  foot. 


Common  blue  stone,  *  160 

Silver-gray  flagging,  -  185 

Stonework,  about,  -  120 

Common  plain  tiles,  -  115 

Sundries. 

Atmospheric  air,  -  0*075 

Yellow  beeswax,  -  -  60 

Birch-charcoal,  -  -  34 

Oak-charcoal,  -  -  21 

Pine-charcoal,  -  -  17 

Solid  gunpowder,  -  -  109 

Shaken  gunpowder,  -  58 

Honey,  -  -  -  90 

Milk/  -  64 

Pitch,  71 

Sea-water,  -  64 

Rain-water,  -  -  -  62*5 

Snow,  -  8 

Wood-ashes,  58 


THE  END. 


BOOKS, 

PUBLISHED  BY 

WILEY  AND  PUTNAM, 


161  Broadway,  HT.  Y. 


K 


% 


I 


DR.  CHEEVER’S  LECTURES  ON  BUNYAN. 

Lectures  on  the  Pilgrim’s  Progress,  and  on  the  Life  and 


Times  of  John  Bunyan.  By  the  Rev.  George 


D.  D.  1  thick  vol.  8vo.,  printed  in  large  type,  with  fine 


steel-plate  engravings.  $3  50 ;  or  in  15  numbers  at  25 


cents  each 


Contents. — 1.  Bunyan  and  his  Times;  2.  Bunyan’s  Tempta¬ 
tions  ;  3.  Bunyan’s  Examination  ;  4.  Bunyan  in  Prison  ;  5.  Provi¬ 
dence,  Grace,  and  Genius  of  Bunyan  ;  6.  City  of  Destruction  and 
Slough  of  Despond  ;  7.  Christian  in  the  house  of  the  Interpreter  ; 
8.  Christian  on  the  Hill  of  Difficulty;  9.  Christian’s  fight  with 
Apollyon  ;  10.  Christian  in  the  Valley  of  the  Shadow  of  Death  ; 
11.  Christian  and  Faithful  in  Vanity  Fair;  12.  Doubting  Castle 
and  Giant  Despair ;  13.  The  Delectable  Mountains  and  En¬ 
chanted  Ground  ;  14.  Land  Beulah  and  the  River  of  Death  ;  15. 
Christiana,  Mercy,  and  the  Children. 

“We  know  of  nothing  in  American  literature  more  likely  to  be  interesting 
and  useful  than  these  lectures.  The  beauty  and  force  of  their  imagery,  the 
poetic  brilliancy  of  their  descriptions,  the  correctness  of  their  sentiments,  and 
the  excellent  spirit  which  pervades  them,  must  make  their  perusal  a  feast  to  all 
of  the  religious  community.” — Tribune. 


II. 

DOWNING’S  COTTAGE  RESIDENCES. 


Designs  for  Cottage  Residences,  adapted  to  North  America, 
including  Elevations  and  Plans  of  the  Buildings,  and  De¬ 
signs  for  Laying  out  Grounds.  By  A.  J.  Downing,  Esq. 
1  vol.  8vo.  with  very  neat  illustrations.  Second  edition, 
revised.  $2  00. 

A  second  edition  of  the  “  Cottage  Residences”  is  just  published,  as  Part  I. ; 
and  it  is  announced  by  the  Author  that  Part  II.,  which  is  in  preparation,  will 
contain  hints  and  designs  for  the  interiors  and  furniture  of  cottages,  as  well  as 
additional  designs  for  farm  buildings. 


One  of  the  leading  reviews  remarked  that  “  the  publication  of  these  works  J 
may  be  considered  an  era  in  the  literature  of  this  country.”  It  is  certainly  true  l 
that  no  works  went  ever  issued  from  the  American  press  which  at  once  exerted  5 
a  more  distinct  and  extended  influence  on  any  subject  than  have  ihese  upon  the  i 
taste  of  our  country.  Since  the  publication  of  the  first  edition  of  the  “  hand-  S 
scape  Gardening,”  the  taste  for  rural  embellishments  has  increased  to  a  surpris-  1 
ing  extent,  and  in  almost  every  instance  this  volume  is  the  text-book  of  the  s 
improver,  and  the  exponent  of  the  more  refined  style  of  arrangement  and  keeping  s 
introduced  into  our  country  residences.  i 

The  “  Cottage  Residences”  seems  to  have  been  equally  well-timed  and  hap-  5 
pily  done.  Country  gentlemen,  no  longer  limited  to  the  meager  designs  of  tin-  s 
educated  carpenters,  are  erecting  agreeable  cottages  in  a  variety  of  styles  suited  s 
to  the  location  or  scenery.  Even  in  the  West  and  South  there  are  already  > 
many  striking  cottages  and  villas  built  wholly,  or  in  part,  from  Mr.  Downing’s  5 
designs  ;  and  in  the  suburbs  of  some  of  the  cities,  most  of  the  new  residences  are  i 


modified  or  moulded  after  the  hints  thrown  out  in  this  work. 


? 


III. 

DOWNING,  ON  LANDSCAPE  GARDENING. 


A  Treatise  on  Landscape  Gardening;  adapted  to  North 
America,  with  a  view  to  the  improvement  of  Country  Re¬ 
sidences.  Comprising  historical  notices,  and  general  prin¬ 
ciples  of  the  art ;  directions  for  laying  out  grounds,  and 
arranging  plantations;  description  and  cultivation  of  hardy 
trees ;  decorative  accompaniments  to  the  house  and  grounds  ; 
formation  of  pieces  of  artificial  water,  flower-gardens,  etc.  ; 
with  remarks  on  Rural  Architecture.  New  edition,  with 
large  additions  and  improvements,  and  many  new  and 
beautiful  illustrations.  By  A.  J.  Downing.  1  large  vol. 
8vo.  $3  50. 

“This  volume,  the  first  American  treatise  on  this  subject,  will  at  once  take 
the  rank  of  the  standard  work.” — Silliman's  Journal. 

“  Downing’s  Landscape  Gardening  is  a  masterly  work  of  its  kind, — more 
especially  considering  that  the  art  is  yet  in  its  infancy  in  America.” — Loudon's 
Gardener's  Magazine. 

“  Nothing  has  been  omitted  that  can  in  the  least  contribute  to  a  full  and  ana¬ 
lytical  development  of  the  subject;  and  he  treats  of  all  in  the  most  lucid  order, 
and  with  much  perspicuity  and  grace  of  diction.” — Democratic  Review. 

“  We  dismiss  this  work  with  much  respect  for  the  taste  and  judgment  of  the 
author,  and  with  full  confidence  that  it  will  exert  a  commanding  influence. 
They  are  valuable  and  instructive,  and  every  man  of  taste,  though  he  may  not 
need,  will  do  well  to  possess  it.” — North  American  Review. 


IV. 

DOWNING’S  FRUITS  OF  AMERICA. 

The  Fruits  and  Fruit  Trees  of  America  ;  or,  the  culture,  pro¬ 
pagation,  and  management,  in  the  garden  and  orchard,  of 
fruit  trees  generally ;  with  descriptions  of  all  the  finest 
varieties  of  fruit,  native  or  foreign,  cultivated  in  the  gardens 
of  this  country.  Illustrated  with  numerous  engravings  and 
outlines  of  fruit.  By  A.  J.  Downing.  1  vol.  12mo.,  (and 
also  8vo. 

***  This  will  be  the  most  complete  work  on  the  subject  ever  published,  and 
will,  it  is  hoped,  supply  a  desideratum  long  felt  by  amateurs  and  cultivators.  ^ 


V. 


LIFE  AND  ELOQUENCE  OF  LARNED. 

Life  and  Eloquence  of  the  Rev.  Sylvester  Larned,  First  Pas¬ 
tor  of  the  First  Presbyterian  Church  in  New  Orleans.  By 
R.  R.  Gurley.  1  thick  vol.  12mo.,  with  a  fine  portrait. 
$1  25. 

Contents. — Preface,  Life  of  Lamed,  Prayer,  Sermons,  Christ 
as  Man,  Paul  before  Felix,  Saving  Faith,  Obligations  for  Spirit¬ 
ual  Mercies,  On  Objections  against  Christianity — the  same,  part 
2 — Practical  Admonitions,  On  the  Inspiration  of  the  Scriptures, 
On  Searching  the  Scriptures,  Religious  Education,  Duty  of  Re¬ 
conciliation  to  God,  Causes  of  Distaste  for  Religion,  Sin  Incon¬ 
sistent  with  Piety,  On  the  Advent,  Walking  in  Wisdom,  Enmity 
of  the  Carnal  Mind,  Duty  to  Orphans,  Excuses  of  the  Impenitent, 
Christian  Self-Examination,  The  Character  of  Herod,  Character 
of  Peter — the  same,  part  2 — Character  of  Paul,  On  the  Resurrec¬ 
tion,  Against  Profane  Swearing,  Love  of  Darkness  rather  than 
Light,  Cause  of  Love  to  God,  Divine  Law  inexorable.  Report  of 
the  Watchman,  Hope  of  the  Righteous,  Moral  Insanity  of  Man. 

“  No  minister  of  the  same  age  has  ever,  at  least  in  this  country,  left  behind 
him  deeper  impressions  of  his  eloquence.  This  volume  is  worthy  of  critical 
examination  and  study ;  and  those  who  would  combine  in  their  sermons  ease 
and  elevation,  simplicity  and  energy;  who  would  leave  to  their  hearers  no  time 
to  sleep,  and  no  wish  to  be  absent,  but  regret  only  at  the  brevity  of  the  service, 
and  delight  at  the  return  of  the  Sabbath,  will  find  the  perusal  and  re-perusal  of 
Mr.  Larned’s  discourses  greatly  to  their  advantage.” — Knickerbocker. 

“  A  beautiful  and  eloquent  tribute  to  sanctified  genius.  The  unity,  force,  ima¬ 
gination,  harmony,  and  feeling  apparent  in  these  discourses,  will  commend  the 
volume  to  all.” — Christian  Observer. 


>  “  A  valuable  treasure  to  all  who  cherish  the  memory  of  one  of  the  most  purc- 

>  minded  and  eloquent  clergymen  of  our  country ;  or  who  know  how  to  appre- 
j  ciate  the  finest  specimens  of  pulpit  composition.” — Tribune. 

>  11  He  was  one  of  the  most  eloquent  orators  in  the  United  States.  Mr.  Gurley 
;  has  made  a  most  interesting  volume,  which  will  prove  an  acceptable  present  to 

>  the  religious  public.” — Evening  Post. 

>  “  A  most  delightful  volume.  We  heartily  commend  it  to  the  religious  com- 
;  munity.” — New  York  .American. 

;  “It  is  much  to  be  wondered  at,  that  no  permanent  memorial  of  this  distin¬ 
guished  divine  has  ever  before  been  given  to  the  world.  The  volume  cannot  fail 
to  be  sought  for  with  great  avidity.” — Daily  .American  Citizen. 

“  These  discourses  evidently  bear  the  impress  of  a  great  mind — not  only  of  an 
exuberant  fancy,  but  of  gigantic  powers  of  comprehension.  We  indeed  rejoice 
that  the  work  has  at  length  appeared. 

“Larned  was  beyond  all  question  the  brightest  star  of  the  American  pulpit, 
during  the  brief  period  in  which  he  lived.  We  are  gratified  to  see  a  memoir 
of  him  so  worthily  constructed,  and  so  rich  in  interesting  material.  The  sermons 
are  pervaded  by  the  living,  breathing  spirit  of  true  genius,  as  well  as  of  evan¬ 
gelical  truth  and  fervent  devotion.” — Albany  Argus. 

St - 


VI. 

TAPPAN’S  ELEMENTS  OF  LOGIC. 

Elements  of  Logic,  together  with  an  introductory  view  of 
Philosophy  in  general,  and  a  Preliminary  View  of  the 
Reason.  One  thick  vol.  12mo.  $1  00. 

Contents  : — 

Part  1. — Introductory  View  of  Philosophy  in  General. 

“  2. — Preliminary  View  of  the  Reason. 

“  3. — Logic  Proper — Book  I.  Primordial  Logic.  II.  In¬ 

ductive  Logic.  III.  Deductive  Logic.  IV. 
Doctrine  of  Evidence. 

“This  is  an  able  and  learned — the  most  able  and  learned  work  which  has 
ever  a|ipeared  on  the  subject  in  this  country.  It  is  written  in  a  simple,  lucid 
style,  and  with  a  great  precision  of  definition  and  distinction.  We  doubt  not  it 
will  be  appreciated  by  learned  men  and  teachers,  and  become  the  standard  work 
in  its  line.” — New  York  Evangelist. 

“The  subject  is  presented,  on  the  whole,  in  a  far  more  original  and  attractive 
form  than  any  treatise  with  which  we  are  acquainted.  The  writer’s  style  is 
characterized  by  a  peculiar  freshness  and  vivacity,  which,  together  with  his 
admirable  arrangement,  relieves  the  subject  of  that  proverbial  tedium  under  the 
imputation  of  which  it  has  always  labored.  This  work  is  finely  adapted  as  a 
Manual  for  schools  and  colleges,  supplying  a  desideratum  which  has  long  been 
felt  to  exist.  The  book  we  decidedly  regard  as  an  honor  to  the  author,  and  an 
honor  to  the  country.” — JSi’cw  World. 

“  We  have  not  been  able  to  examine  this  excellent  treatise  with  the  attention 
it  merits ;  but  we  think  we  are  safe  in  saying  that  it  is  not  only  the  most  original, 
but  the  best  work  on  Logic,  which  has  ever  appeared  in  this  country.” — Journal 
of  Commerce. 

“  On  the  whole  W'e  think  this  is  the  best  work  on  Logic  which  we  have  seen 
from  the  American  press.” — Evening  Post. 

BY  THE  SAME  AUTHOR. 

Tappan  on  The  Will.  3  vols.  $3  00  ;  or  separately. 

Vol.  1. — Review  of  Edwards. 

“  2. — Appeal  to  Consciousness. 

“  3. — Moral  Agency. 


VII. 

BRADFORD’S  AMERICAN  ANTIQUITIES. 

American  Antiquities,  and  Researches  into  the  Origin  and 
History  of  the  Red  Race.  By  Alexander  W.  Bradford. 
1  vol.  8vo.  $1  00. 

***  A  philosophical  and  elaborate  investigation  of  a  subject  which  lias  excited 
much  attention.  This  able  work  is  a  very  desirable  companion  to  those  of  Ste¬ 
phens  and  others  on  the  Ruins  of  Central  America. 


VIII. 


GRAY’S  BOTANICAL  TEXT  BOOK. 

The  Botanical  Text  Book  for  Colleges,  Schools,  and  private 
Students.  Comprising  not  only  the  outlines  of  Structural 
and  Physiological  Botany,  but  also  a  popular  account  of  the 
principal  Natural  Orders,  their  geographical  distribution, 
properties,  and  uses,  with  an  enumeration  of  those  plants 
which  furnish  products  employed  in  medicine  and  the  arts. 
1  very  thick  vol.  with  numerous  fine  engravings.  $1  50. 

Contents. — Preliminary  Considerations.  Part  I.  Structural 
and  Physiological  Botany.  Part  II.  Systematic  Botany.  Ap¬ 
pendix,  Index,  Glossary  of  Botanical  Terms.  Index  of  the  Na¬ 
tural  Orders,  Useful  Plants,  and  Products,  &c. 

“The  most  compendious  and  satisfactory  view  of  the  Vegetable  Kingdom 
which  has  yet  been  offered  in  an  elementary  treatise.  Remarkable  for  its  cor¬ 
rectness  and  perspicuity.” — Silliman's  Journal. 

See  also  Loudon,  Hooker,  and  other  English  Botanical  Journals,  &c. 


IX. 

NEW  SERIES  OF  THE  BIBLIOTHECA  SACRA. 


BIBLIOTHECA  SACRA, 


AND 

THEOLOGICAL  REVIEW. 

Conducted  by  B.  B.  Edwards  and  E.  A.  Park,  Professors  at 
Andover.  With  the  special  co-operation  of  Dr.  Robinson 
and  Professor  Stuart.  Price  $4  00  a  year. 

“  A  noble  contribution  to  Religious  Literature,  and  fitly  printed.” — Tribune. 

“  Confessedly  one  of  the  ablest  and  most  important  Theological  Reviews  pub¬ 
lished  in  this  country.” — Courier  and  Enquirer. 

“  As  an  aid  to  the  Biblical  Student,  this  is  doubtless  the  most  valuable  peri¬ 
odical  in  the  English  language.  The  other  religions  publications  in  this  coun¬ 
try,  admitting  a  wider  range  of  subjects,  cannot  concentrate  so  much  strength 
on  the  department  of  Biblical  learning.  None  of  them  therefore  can  adequately 
supply  its  place;  but  the  principal  recommendation  of  this  work,  after  all,  is  its 
elevated  and  manly  tone.” — jYcw  York  Observer. 

“This  is,  perhaps,  the  most  ambitious  journal  in  the  United  States.  We  use 
the  word  in  a  good  sense,  as  meaning  that  there  is  no  journal  among  us  which 
seems  more  laudably  desirous  to  take  the  lead  in  literary  and  theological  science. 
Its  handsome  type  and  paper  give  it  a  pleasing  exterior ;  its  typographical  errors, 
are  so  comparatively  few,  as  to  show  that  it  has  the  advantage  of  the  best 
American  proof-reading;  while  for  thoroughness  of  execution  in  the  depart¬ 
ments  of  history  and  criticism,  it  aims  to  be  pre-eminent.” — Churchman. 


X. 

GARDENING  FOR  LADIES 


\  Gardening  for  Ladies  ;  and  Companion  to  the  Flower-Garden. 
|  Being  an  Alphabetical  arrangement  of  all  the  ornamental 
'>  Plants  usually  grown  in  gardens  and  shrubberies ;  with 

>  full  directions  for  their  culture.  By  Mrs.  Loudon.  First 
\  American,  from  the  second  London  edition.  Revised  and 

>  edited  by  A.  J.  Downing.  1  thick  vol.  12mo.,  with  en- 

\  gravings  representing  the  processes  of  grafting,  budding, 
•  layering,  &c.,  &c.  $1  50. 

1“  A  truly  charming  work,  written  with  simplicity  anti  clearness.  It  is  deci¬ 
dedly  the  best  work  on  the  subject,  and  we  strongly  recommend  it  to  all  our 
fair  countrywomen,  as  a  work  they  ought  not  to  be  without.” — N.  Y.  Courier. 

“  Mr.  Downing  is  entitled  to  the  thanks  of  the  fair  florists  of  our  country  for 
introducing  to  their  acquaintance  this  comprehensive  and  excellent  manual, 
i  which  must  become  very  popular.  Besides  an  instructive  treatment  on  the  best 
<  modes  of  culture,  transplanting,  bedding,  training,  destroying  insects,  &c.,  and 
j  the  management  of  plants  in  pots  and  green-houses,  illustrated  with  numerous 
<  plates;  the  work  comprises  a  Dictionary  of  the  English  and  Botanic  names  of 
s  the  most  popular  flowers,  with  directions  for  their  culture.  Altogether  we 
<  should  judge  it  to  be  the  most  valuable  work  in  the  department  to  which  it 
;  belongs.” — Newark  Advertiser. 

<  “This  is  a  full  and  complete  manual  of  instruction  upon  the  subject  of  which 

Iit  treats.  Being  intended  for  those  who  have  little  or  no  previous  knowledge  of 
gardening,  it  presents,  in  a  very  precise  and  detailed  manner,  all  that  is  neces¬ 
sary  to  be  known  upon  it,  and  cannot  fail  to  awaken  a  more  general  taste  for 
these  healthful  and  pleasant  pursuits  among  the  ladies  of  our  country.” — N.  Y. 
\  Tribune. 

<  “This  truly  delightful  work  cannot  be  too  highly  commended  to  our  fair  coun- 
l  try  women.” — N.  Y.  Journal  of  Commerce. 

t  “We  cordially  welcome,  and  heartily  commend  to  all  our  fair  friends,  whether 
<  living  in  town  or  country,  this  very  excellent  work.” — N.  Y.  Tribune. 


XI. 

THE  BIRDS  OF  LONG  ISLAND- 

i  Containing  a  description  of  the  habits,  plumage,  &c.,  of  all 
\  the  species  now  known  to  visit  that  section,  comprising  the 
{  larger  number  of  birds  found  throughout  the  State  of  New 
j  York,  and  the  neighboring  States.  By  T.  P.  Giraud,  jr. 
ij  1  vol.  8vo.  Price  $2  00. 

/  This  work,  though  designed  chiefly  for  the  use  of  the  gunners  and  sportsmen 
\  residing  on  Long  Island,  will  still  serve  as  a  hook  of  reference  for  amateurs  and 
>  others  collecting  ornithological  specimens  in  various  sections  of  the  United 
I:  States,  particularly  for  those  persons  residing  on  the  sea-coasts  of  New  Jersey 
$  and  the  Eastern  States. 


XII. 


LINDLEY  ON  HORTICULTURE. 

The  Theory  of  Horticulture ;  or  an  attempt  to  explain  the 
principal  operations  of  gardening  upon  physiological  prin¬ 
ciples.  By  John  Lindley,  Ph.  D.,  F.  R.  S.,  with  notes 
and  additions  by  A.  J.  Downing,  and  Dr.  A.  Gray.  1 
thick  vol.  12mo.,  with  engravings.  $1  25. 

Contents. — Of  Germination,  Of  growth  by  the  root,  Growth 
by  the  Stem,  Action  of  Leaves,  Action  of  Flowers,  Of  the  matu¬ 
ration  of  the  Fruit,  Of  Temperature,  Of  Bottom-heat,  Moisture  of 
the  Soil,  Watering,  Atmospherical  Moisture  and  Temperature, 
Ventilation,  Seed-sowing,  Seed-saving,  Seed-packing,  Propagation 
by  Eyes  and  Knaws,  By  Leaves,  By  Cuttings,  By  Layers  and 
Suckers,  By  Budding  and  Grafting,  Of  Pruning,  Training,  Pot¬ 
ting,  Transplanting,  Of  the  preservation  of  races  by  Seed,  Of  the 
improvement  of  Races,  Of  Resting,  Of  Soil  and  Manure,  Index. 

“A  vast  fund  of  horticultural  learning,  and  embraces,  it  is  hardly  too  much  to 
say,  nearly  all  that  an  intelligent  gardener  need  know.” — Loudon's  Magazine  of 
Gardening. 

“  We  are  constrained  to  believe  that  it  will  provide  the  intelligent  gardener 
and  the  scientific  amateur  with  correct  means  of  learning  the  more  important 
operations  of  horticulture.” — Farmer's  Magazine. 

“The  American  edition  of  this  valuable  work  is,  in  all  respects,  creditable  to 
the  editors;  whose  joint  labors,  it  may  be  remarked,  furnish  in  the  present  in¬ 
stance  another  illustration  of  the  happy  combination  of  scientific  theory  with 
practical  experience.  To  the  American  reader,  the  notes  of  the  co-editors, 
which  are  both  scientifical  and  practical,  add  much  to  the  value  and  interest  of 
the  work  ;  being,  for  the  most  part,  the  results  of  successful  experience,  with 
such  additions  and  adaptations  as  the  climate  and  circumstances  of  our  country 
render  necessary.” — American  Journal  of  Science. 


XIII. 

THE  CROTON  AQUEDUCT. 

Illustrations  of  the  Croton  Aqueduct.  By  F.  B.  Tower,  of 
the  Engineer  Department.  1  handsome  vol.  4to.,  with  25 
line  engravings.  $3  50. 

“This  volume  is  very  elegant,  and  must  be  extremely  popular  as  a  permanent 
and  beautiful  record  of  one  of  the  greatest  works  of  modem  times.” — JV*.  Y. 
Tribune. 

“  Here  is  a  book  which  every  New  Yorker  ought  to  buy  who  has  means  to 
have  a  library,  and  can  afford  to  pay  the  price  of  it,  without  actually  depriving 
himself  of  necessities,  and  out  of  New  York  everybody  ought  to  buy  it  who  is 
able  to  indulge  a  taste  for  elegant  and  valuable  books.” — JV*.  Y.  Commercial.  jf 


ixiv. 

JOHNSTON’S  AGRICULTURE. 

Lectures  on  the  Application  of  Chemistry  and  Geology  to 
Agriculture.  By  J.  F.  W.  Johnston.  Complete  in  one 
thick  vol.  $1  25;  or  in  2  vols.  $1  50. 

Contents  : — 

Part  1. — On  the  Organic  Constituents  of  Plants. 

“  2. — On  the  Inorganic  Constituents  of  Plants. 

1“  3. — On  the  Improvement  of  the  Soil  by  Mechanical 
and  Chemical  means. 

“  4. — On  the  Products  of  the  Soil  and  their  use  in  the 
Feeding  of  Animals. 

Appendix. — Of  Suggestions  and  Results  of  Experiments  in 
/  Practical  Agriculture. 

i  “It  is  unquestionably  the  most  important  contribution  to  agricultural  science, 
<  and  destined  to  exert  a  most  beneficial  influence  in  this  country.” — Professor 
<  Silliman. 

£  “A  work  of  great  value  to  the  agriculturist  who  would  avail  himself  of  the 
s  aid  of  science  in  the  cultivation  of  his  land.” — Am.  Agriculturist. 

<;  “This  truly  valuable  work  forms  the  only  complete  treatise  on  the  whole 
<  subject  to  be  found  in  any  language.” — Blackwood' s  Magazine. 

I  “The  most  complete  account  of  Agricultural  Chemistry  we  possess.” — Royal 
'  Agricultural  Journal. 

(  “We  only  wish  it  were  in  the  hands  of  every  farmer’s  son  in  the  country.” — 
<  Durham  Advertiser. 

<  “  Nothing  hitherto  published  has  at  all  equalled  it,  both  as  regards  true  science 

\  and  sound  common  sense.” — Quar.  Journal  of  Agriculture. 

i  “A  valuable  and  interesting  Course  of  Lectures.” — London  Quar.  Review. 

XV. 

WATER  CURE,  FOR  LADIES. 

>  A  popular  work  ou  the  Health,  Diet,  and  Regimen  of  Fe¬ 
ll  males  and  Children,  and  prevention  and  cure  of  diseases ; 

>  with  a  full  account  of  the  process  of  Water  Cure,  illustrated 

l  with  various  cases,  by  Mrs.  M.  L.  Shew,  revised  by  Joel 

;  Shew,  M.  D.  1  vol.  Price  50  cents. 

I  u  A  valuable  and  instructive  work  on  that  most  interesting  branch  of  modem 
>  medical  science,  the  medical  virtues  of  water.”— JV.  Y.  Express. 

\  “The  authoress  has  reduced  the  system  to  practice,  and  found  it  every  way 
l  equal  in  its  curative  influences  to  the  representations  of  its  many  advocates.” — 
<  True  Sun. 


XVI. 

ACTONI AN  PRIZE  ESSAY. 

Chemistry,  as  exemplifying  the  Wisdom  and  Beneficence  of 
God.  By  George  Fownes,  Ph.  D.,  F.  R.  S.,  Etc.  In  1 
vol.  small  8vo.  Price  50  cents. 

Contents. — The  Chemical  History  of  the  Earth  and  the  At¬ 
mosphere  ;  The  Peculiarities  which  characterize  Organic  Sub¬ 
stances  generally  ;  The  Composition  and  Sustenance  of  Plants  ; 
On  Animal  Chemistry  ;  The  Relation  existing  between  Plants 
and  Animals  ;  Appendix — (with  various  Tables.) 

“Tlie  object  of  the  work  is  to  gather  up  the  proofs  ami  indications  of  design 
and  goodness  in  the  structure  and  relations  of  tilings  disclosed  by  Chemistry — 
and  it  is  very  ably  done.” — JV.  Y.  Post. 

“It  is  richly  worth  general  perusal.” — JV.  Y.  Tribune. 

“The  manner  of  treating  the  subject  is  both  ingenious  and  recondite,  and  we 
commend  it  accordingly  to  general  attention.” — JV.  Y.  American. 

“  A  highly  interesting  and  valuablp  work.  It  is  a  most  valuable  addendum  to 
other  works  on  this  subject;  to  those  who  are  studying  Natural  Theology,  it 
will  be  highly  serviceable.” — JV.  Y.  Express. 

“This  is  a  meritorious  work.  The  materials  are  fairly  and  skilfully  selected 
out  of  the  vast  and  ever-growing  mass  of  phenomena  and  truths  which  consti¬ 
tute  the  modern  science  of  Chemistry ;  and  are  pm  together  with  considerable 
dexterity,  imparting  an  air  of  novelty  and  frcshuess  even  to  the  truths  with 
which  we  have  been  long  familiar.” — Christian  Remembrancer. 


XVII. 

HOLY  BIBLE,  WITH  COMMENTARY. 

Now  ready — Vols.  1  and  2,  $4  00  each;  or,  numbers  1  to  28, 
of  the  Holy  Bible,  with  a  Critical  Commentary  and  Para¬ 
phrase,  by  Patrick,  Lowth,  Arnald,  Whitby,  and  Lowman. 
A  new  edition,  with  the  text  printed  at  large.  To  be  com¬ 
pleted  in  sixty  numbers,  at  25  cents  each,  the  whole  to  form 
four  imperial  octavo  volumes,  containing  upwards  of  4,300 
pages.  The  value  of  this  edition  consists  in  the  fact  that 
the  Text  accompanies  the  Commentaries — thus  adapting  it 
to  general  use.  » 

Students,  Clergymen,  and  others  clubbing  together,  and  remitting  the 
Publishers  the  amount  of  live  copies,  will  be  entitled  to  the  sixth  gratis;  or 
twelve  copies  for  ten,  and  in  the  same  proportion  for  a  larger  number. 

The  whole  cost  of  the  publication  is  not  required  in  advance,  as  the  work 
)  can  be  forwarded  in  either  numbers  or  volumes,  as  the  parly  may  desire. 


\ 


sr 


XVIII. 

DANA’S  MINERALOGY. 


A  System  of  Mineralogy  ;  Comprising  the  most  recent  dis¬ 
coveries,  with  numerous  engravings.  Second  edition, 
enlarged  and  improved.  By  James  D.  Dana,  A.  M. 

Very  thick  vol.  8vo.,  pp.  633.  $3  50. 

Contents.  —  Introduction.  Part  I.  Crystallogony,  or  the 
Science  of  the  Structure  of  Minerals.  II.  Physical  Properties 
of  Minerals.  III.  Chemical  Properties  of  Minerals.  IV.  Taxo¬ 
nomy.  V.  Determinative  Mineralogy.  VI.  Descriptive  Minera¬ 
logy.  VII.  Chemical  Classification.  VIII.  Rocks  on  Mineral 
Aggregates.  IX.  Mineralogical  Bibliography.  X.  Copious  Index. 

“  It  gives  me  great  pleasure  to  state  that  it  requires  but  few  works  like  the 
present,  to  give  American  Science  a  name  which  will  merit,  if  it  does  not  re¬ 
ceive,  the  respect  of  tile  scientific  world.” — Silliman’s  Journal  for  April. 

“This  work  does  great  honor  to  America,  and  should  make  11s  blush  for  the 
neglect  in  England  of  an  important  and  interesting  science.  It  is  a  thick  octavo, 
of  about  700  pages,  on  Mineralogy,  treated  in  a  highly  scientific  and  perspicuous 
manner.  It  is  no  compilation,  such  as  all  works  on  this  subject  have  been  in 
this  country  since  the  writings  of  Jameson  and  Phillips,  but  an  original  survey 
of  the  mineral  kingdom  executed  with  the  greatest  care.  This,  too,  is  the  second 
edition,  greatly  enlarged,  showing  that  Mr.  Dana’s  labors  are  appreciated  in 
America.” — London  Athenceum. 

“  This  work  bears  marks  on  every  page  of  great  industry  and  determination 
in  collecting  the  most  recent  facts.  In  completeness,  systematic  arrangement, 
and  accuracy,  it  is  believed  to  be  exceeded  by  no  other  work  extant.” — JV.  Y. 
American. 

“This  is  a  new  edition  of  the  best  treatise  ever  published  in  this  country  on 
the  interesting  and  important  subject  of  Mineralogy,  it  first  appeared  seven 
years  ago,  since  which  time  many  new  discoveries  have  been  made  in  the 
science,  and  sources  have  thus  been  opened  for  a  vast  amount  of  new  .and  im¬ 
portant  matter.  All  the  investigations,  both  Foreign  and  American,  that  have 
been  made,  have  been  carefully  consulted  in  the  preparation  of  this  new  edition, 
and  a  chapter  on  crystallography  has  been  added.  The  work  is  a  most  welcome 
addition  to  the  series  of  American  standard  treatises  on  scientific  subjects.” — JV. 
Y.  Courier  and  Enquirer. 

“This  is  a  truly  valuable  and  learned  work,  and  it  is  surprising,  considering 
the  correctness  of  this  treatise  on  its  first  appearance,  to  find  how  numerous  and 
important  are  the  changes  which  have  been  made  in  the  present  edition.  We 
are  sure  the  work  must  command  success.” — Tribune. 


XIX. 

HAND-BOOK  OF  NEEDLEWORK. 

The  Hand-Book  of  Needle  Work.  By  Miss  Lambert.  1 
vol.  8vo.,  beautifully  printed,  with  numerous  illustrations. 
Price  $1  50  ;  or  in  extra  binding,  neat  fancy  style,  $3  00. 

This  very  elegant  and  useful  volume  proves  to  be  the  most  attractive  work 
of  the  kind  ever  published  in  this  country.  It  contains  practical  instructions  in 
the  various  kinds  of  Ornamental  Needlework  and  Embroidery,  with  a  historical 
account  of  these  accomplishments  in  all  ages  and  nations.  To  use  a  common 
phrase,  it  certainly  deserves  a  place  on  every  lady’s  work  table,  besides  being  an 
ornament  to  the  drawing-room. 


XX. 

LETTERS  AND  DESPATCHES  OF  CORTES, 


% 


The  Despatches  of  Fernando  Cortes,  the  Conqueror  of  Mexi¬ 
co,  addressed  to  the  Emperor  Charles  V. ;  written  during 
the  Conquest,  and  containing  a  narrative  of  its  events. 
Translated  by  George  Folsom,  Secretary  of  the  N.  Y. 
Historical  Society.  In  1  vol.  Med.  8vo.  $1  25.  Large 
Paper  copies,  $2  00. 

“  We  venture  to  pronounce  this  one  of  the  most  curious  and  most  interesting 
books  that  have  made  their  appearance  for  some  time.  These  despatches  have 
never  before  been  seen  in  the  English  language,  and  one  of  them  at  least  has 
never  been  printed  even  in  Spain.  The  very  title  is  enough  to  arouse  a  deep 
interest.  The  Conquest  of  Mexico,  written  by  the  Conqueror  himself,  on  the 
very  field  of  battle!  We  can  scarcely  think  of  a  rarer  desideratum.” — JV.  Y. 
Courier. 

“These  very  interesting  records  of  a  National  Military  Romance,  which 
created  a  new  world,  and  produced  most  marvellous  changes  by  its  influence  on 
the  old.  The  translation  is  ably  performed.” — Literary  Gazette. 

“This  is  a  volume  which  ought  to  find  a  niche  in  every  well  furnished  libra¬ 
ry.  It  presents  a  most  extraordinary  autograph  picture  by  one  of  the  most  extra¬ 
ordinary  characters  of  our  modern  history.” — Globe. 

“This  book  is  a  credit  to  the  American  press.  The  Despatches  of  Cortes  are 
among  the  most  interesting  and  singular  documents  ever  penned.  They  give  a 
minute  and  vivid  account  of  his  conquest,  and  of  the  wonderful  scenes  presented 
to  his  view  on  his  first  entry  into  the  Kingdom  of  Mexico.” — Britannia. 

“This  book  has  all  the  interest  of  a  novel,  and  all  the  value  of  a  history. 
What  higher  praise  can  we  bestow  upon  such  a  work  1  This  marvellous,  this 
unparalleled  story.” — Tablet. 

“He  preserves  an  interest  in  his  narrative  read  even  at  this  distance,  when 
the  mysterious  novelties  of  the  country,  the  importance  of  the  facts,  and  the 
uncertainty  of  the  result,  have  long  ceased  to  impart  an  interest.” — Spectator. 

“This  is  one  of  the  most  curious  publications  of  the  day.  A  valuable  histori¬ 
cal  document,  containing  an  exact  and  picturesque  representation  of  the  habits 
and  manners  of  a  people  long  since  extinct.” — Bell's  Weekly  Messenger. 


XXI. 

THE  YOUNG  AMERICAN’S  LIBRARY,  NO.  1. 


THE  PRIMER: 

With  over  200  neat  engravings,  most  beautifully  printed,  in 
quite  a  new  and  novel  style.  Price  25  cents. 

“  As  pretty  a  little  book  for  little  people  as  we  ever  saw.  It  is  full  of  beauti¬ 
ful  pictures,  which  convey  some  useful  lessons  to  the  child  while  he  is  thinking  < 
of  nothing  but  pleasure.  I(  strikes  the  great  secret  of  education.  The  getting  > 
up  of  this  book  is  unusually  fine,  and  we  learn  it  is  the  first  number  of  a  series  $ 
corresponding  to  the  name.”— JV.  Y.  Tribune.  $ 


XXII. 


NOTES  ON  NORTHERN  AFRICA. 


>,  Notes  on  Northern  Africa,  the  Sahara  and  Soudan,  in  rela- 

|  tion  to  the  Ethnography,  Languages,  History,  Political 

i  and  Social  Condition  of  the  nations  of  those  countries, 

5  with  various  vocabularies.  By  William  B.  Hodgson.  1 

'■  vol.  8vo.,  well  printed.  Price  75  cents. 

>  Contents. — Barbary,  Kabyles,  Tuarycks,  Mozabees,  Wurge- 
lans,  Wadreagans,  Sergoos,  Siwahees,  Schelouh,  Guanches,  Nu- 
midian  Inscription  in  America,  Foolalis  or  Fellatahs,  Tibbos, 
Bornouees,  Haoussans,  Timbuctoo. 

***  The  Information  contained  in  these  interesting  pages,  is  the  result  of  the 
author's  personal  intercourse  with  the  natives  of  Africa.  During  his  official 
residence  at  Algiers,  he  had  opportunities  of  conversing  with  persons,  from  the 
various  countries  which  he  has  described.  W hat  he  has  related,  was  repeatedly 
confirmed  by  successive  inquiries.  The  facts  recorded  may.  therefore,  he  deemed 
as  near  an  approximation  to  truth,  as  the  circumstances  of  the  case  would  allow. 
There  was  no  other  mode,  at  least,  of  obtaining  information  so  important  to 
science;  as  no  European  has  yet  visited  that  region  of  Africa,  which  lies  im¬ 
mediately  south  of  Algiers.  With  the  hope  that  these  notes  may  afford  some 
additional  light  upon  the  obscure  history  of  Africa,  and  that  interesting  portion 
of  the  human  race  they  are  now  published. 


XXIII. 

NORDHEIMER’S  HEBREW  GRAMMAR. 

A  Critical  Grammar  of  the  Hebrew  Language,  by  Isaac 
Nordheimer,  Ph.  D.,  late  Professor  of  Arabic  and  other 
Oriental  Languages  in  the  University  of  New  York.  3 
vols.  8vo.,  including  Chrestomathy.  $7  00. 

“Ilis  first  volume  was  most  favorably  noticed  by  several  periodicals,  both  at 
home  and  abroad.  The  second  has  even  a  higher  claim  to  commendation,  not 
only  for  the  great  beauty  and  neatness  of  its  execution,  but  still  more  for  the 
perspicuity  of  its  style,  and  the  intrinsic  excellence  of  its  matter.” — Biblical 
Repository. 

“To  clergymen  and  others,  who  would  be  glad  to  recover  and  increase  their 
knowledge  of  the  Hebrew,  an  attentive  study  of  this  work  would  afford  an 
invaluable  aid,  and  we  may  add,  delightful  entertainment.” — Princeton  Review. 

“The  delightful  ease  with  which  we  pass  over  its  pages,  the  interesting  man¬ 
ner  in  which  the  author  has  laid  open  to  us  the  processes  of  our  own  minds,  the 
many  apposite  and  beautiful  examples  adduced  by  way  of  illustration,  the  ab¬ 
sence  of  all  pedantry,  its  freedom  from  far-fetched  theorizing  and  illogical  reason¬ 
ing,  produce  such  an  impression  of  ease,  truth,  and  clearness,  that  we  almost  claim 
the  thoughts  and  conclusions  as  our  own,  so  spontaneously  do  our  minds  meet 
those  views  which  are  everywhere  presented.” — Report,  of  Bibl.  Lit. 

“We  are  free  to  say  that,  as  a  whole,  the  exhibition  of  the  Hebrew  Language 
in  its  peculiar  idioms,  the  arrangement  of  those  idioms  on  a  systematic  plan, 
and  the  solution  of  them  by  a  reference  to  universal  principles,  have  been  ac¬ 
complished  in  a  manner  eminently  able  and  successful.” — Eclectic  Review. 


Manners  and  Customs  of  the  North  American  Indians.  In  < 
Letters  and  Notes  written  during  eight  years  travel  among  > 
the  wildest  tribes  of  Indians  in  North  America,  with  400  > 
spirited  illustrations,  carefully  engraved  from  his  Original  s 
Paintings.  By  George  Catlin.  A  new  edition  in  2  vols.  ? 
royal  8vo.  Price  $6  00,  bound  in  cloth. 

***  Four  editions  of  this  very  interesting  work  have  been  printed  in  London.  < 
Among  the  subscribers  were  the  Queen,  the  Queen  Dowager,  the  King  of  Bel-  < 
gium,  and  many  of  the  most  distinguished  persons  in  Europe.  It  contains  char-  < 
acteristic  and  faithful  records  of  a  race  of  people  who  are  rapidly  becoming  ex-  < 
ti net :  and  it  is  not  probable  that  another  similar  work  can  ever  be  written.  < 
One  of  the  most  remarkable  tribes,  the  Mandans,  are  already  entirely  destroyed.  \ 


XXVII. 

HAND-BOOK  OF  HYDROPATHY. 

Hand-Book  of  Hydropathy  ;  or  a  Popular  Account  of  the 
Treatment  and  Prevention  of  Diseases,  by  means  of  Wa¬ 
ter.  Chiefly  selected  from  the  most  eminent  and  recent 
European  authors,  by  Joel  Shew,  M.  D.  1  vol.  12mo. 
Second  edition.  Price  50  cents  ;  or  in  paper  binding,  38  cts. 

“This  excellent  little  work  of  Dr.  Shew  has  been  compiled  from  the  best  au¬ 
thors,  and  contains  as  complete  a  view  of  the  practice  under  the  mode  as  can  be 
given.” — JV.  T.  Pust. 

“It  is  eminently  calculated  to  benefit  all  who  read  and  study  it,  whether  sick 
or  well.” — Regenerator. 

“  This  book  is  well  printed,  its  contents  have  been  judiciously  selected  from 
a  variety  of  sources,  and  it  gives  a  complete  compend  of  the  Treatment  by  Water 
in  its  present  state  of  improvement.  It  is  universally  calculated  to  do  good  in 
the  all-important  matter  of  preventing ,  as  well  as  curing  disease.” — JV.  T. 
Tribune. 


XXVIII. 

LOCKHART’S  SPANISH  BALLADS. 

Ancient  Spanish  Ballads,  Historical  and  Romantic,  translated, 
with  notes,  by  J.  G.  Lockhart,  Esq.  To  which  are  added, 
an  Essay  on  the  Origin,  Antiquity,  Character,  and  Influ¬ 
ence  of  the  Ancient  Ballads  of  Spain  ;  and  an  Analytical 
Account,  with  Specimens,  of  the  Romance  of  the  Cid.  1 
very  neat  vol.  8vo.,  beautifully  printed.  $1  50. 

“These  ‘Spanish  Ballads’  are  known  to  our  public,  but  generally  with  incon¬ 
ceivable  advantage,  by  the  very  tine  and  animated  translations  of  Mr.  Lock¬ 
hart.” — Hallam. 

“  This  delightful  volume  needs  no  commendation  of  ours ;  every  one  will  buy 
it,  arid  keep  it  among  their  literary  treasures.” — Edinburgh  Review. 

“  We  are  quite  at  a  loss  to  speak  in  adequate  terms  of  this  delightful  and  in¬ 
teresting  volume,  the  perusal  and  reperusal  of  which  have  afforded  us  so  much 
real  gratification, — but  we  advise  every  one  to  get  it.” — JV.  Tribune. 


XXIX. 

NEW  TABLES  OF  INTEREST. 

Tables  of  Interest,  determining,  by  means  of  the  Differences 
of  Logistic  Squares,  the  interest  of  every  whole  sum  up  to 
10,000  dollars,  for  any  length  of  time  not  exceeding  400 


days,  at  the  rates  of  6  and  7  per  cent, 
beautifully  printed.  $1  50. 


1  vol.  royal  8vo., 


“The  application  of  the  tables  appears  to  be  so  direct  and  plain,  and  the 
method  of  using  them  so  concise,  that  we  can  safely  recommend  the  book  as 
worthy  of  adoption  among  merchants,  bankers,  and  others.” — JV.  Y.  Commercial 
Advertiser. 

“The  very  slight  amount  of  numerical  calculation  required  in  using  these 
tables  and  the  uniformity  of  the  process  appear  to  give  the  work  a  claim  on  the 
attention  of  those  whose  business  requires  tire  frequent  computation  of  interest.” 
— JV.  Y.  Post. 

“This  work  seems  to  answer  fully  the  purpose  for  which  it  was  prepared, 
in  furnishing  to  the  business  community  a  concise  and  easy  method  of  finding 
the  interest  of  money.” — JV.  Y.  American. 


XXX. 

SHORT  AND  SIMPLE  PRAYERS 


,  HYMNS  FOR  THE  USE  OF  CHILDREN. 

By  the  Author  of  “  Mamma’s  Bible  Stories.”  1  vol.,  with 

>  neat  engravings.  Price  37  cents. 

? 

“Prayer  is  the  simplest  form  of  speech 
That  infant  lips  can  try.” — Montgomery. 

“We  do  not  pretend  to  remember  the  many  little  books  similar  in  design  to 
this  which  we  may  have  received,  but  none  that  we  can  recall  seems  so  well 
adapted  to  its  purpose.  The  prayers  and  hymns  are  peculiarly  simple  and 
touching.  The  heart  of  a  child  could  hardly  fail  to  be  moved  by  them.  The 
volume  is  a  neat  one,  very  well  printed,  with  two  or  three  pretty  illustrations.” — 
North  American. 


XXXI. 

HAPPY  HOURS, 

OR,  THE  HOME  STORY-BOOK. 

By  Mary  Cherwell.  1  vol.  with  neat  engravings,  handsomely 
printed  in  large  bold  type.  Second  edition.  Price  50  cts. 

“  A  sweet  little  book  of  home  stories,  which  all  young  people  will  be  delighted 
with.” — JV.  Y.  Tribune. 


?  “We  can  scarcely  commend  this  little  book  enough;  the  enterprising  pub- 
>  lishers  are  entitled  to  great  praise  for  the  handsome  style  in  which  it  is  pub- 
.'  lished.” — True  Sun. 

s 

5  “  A  delightful  book  for  children  :  it  is  very  pleasantly  written,  and  cannot  fail 

$  to  engage  the  young  reader’s  attention.  The  designs  are  pretty,  and  neatly  ex¬ 
ecuted.  We  strongly  recommend  it  to  all  our  young  friends.” — JV.  Y.  Express. 


% 


GETTY  CENTER  LIBRARY 


